surface+area

=media type="custom" key="12091711"= **__ Surface Area and Volume __** = = // This chapter extends what we know about two-dimensional figures to three-dimensional shapes. Different types of 3D shapes and their parts are defined. Then, surface area and volume of prisms, cylinders, pyramids, cones, and spheres are found. // = = = = In this chapter we extend what we know about two-dimensional figures to three-dimensional shapes. First, we will define the different types of 3D shapes and their parts. Then, we will find the surface area and volume of prisms, cylinders, pyramids, cones, and spheres.

click on this link to manipulate an area while you determine volum

Learning Objectives

 * Identify different types of solids and their parts.
 * Use Euler’s formula and nets.

Review Queue

 * 1) Draw an octagon and identify the edges and vertices of the octagon. How many of each are there?
 * 2) Find the area of a square with 5 cm sides.
 * 3) Draw the following polygons.
 * 4) A convex pentagon.
 * 5) A concave nonagon.
 * Know What?** Until now, we have only talked about two-dimensional, or flat, shapes. Copy the equilateral triangle to the right onto a piece of paper and cut it out. Fold on the dotted lines. What shape do these four equilateral triangles make?

Polyhedrons
Each polygon in a polyhedron is a **//face.//** The line segment where two faces intersect is an **//edge.//** The point of intersection of two edges is a **//vertex.//** Examples of polyhedrons include a cube, prism, or pyramid. Non-polyhedrons are cones, spheres, and cylinders because they have sides that are not polygons. All prisms and pyramids are named by their bases. So, the first prism would be a triangular prism and the first pyramid would be a hexagonal pyramid. a) b) c) a) The base is a triangle and all the sides are triangles, so this is a triangular pyramid. There are 4 faces, 6 edges and 4 vertices. b) This solid is also a polyhedron. The bases are both pentagons, so it is a pentagonal prism. There are 7 faces, 15 edges, and 10 vertices. c) The bases that are circles. Circles are not polygons, so it is not a polyhedron.
 * Polyhedron:** A 3-dimensional figure that is formed by polygons that enclose a region in space.
 * Prism:** A polyhedron with two congruent bases, in parallel planes, and the lateral sides are rectangles.
 * Pyramid:** A polyhedron with one base and all the lateral sides meet at a common vertex.
 * Example 1:** Determine if the following solids are polyhedrons. If the solid is a polyhedron, name it and find the number of faces, edges and vertices each has.
 * Solution:**

Euler’s Theorem
Let’s put our results from Example 1 into a table. Notice that faces + vertices is two more that the number of edges. This is called Euler’s Theorem, after the Swiss mathematician Leonhard Euler. Because the two sides are not equal, this figure is not a polyhedron.
 * || **//Faces//** || **//Vertices//** || **//Edges//** ||
 * **//Triangular Pyramid//** || 4 || 4 || 6 ||
 * **//Pentagonal Prism//** || 7 || 10 || 15 ||
 * Euler’s Theorem:** [[image:http://www.ck12.org/ck12/ucs/?math=F%2BV%3DE%2B2 caption="F+V=E+2"]].
 * Example 2:** Find the number of faces, vertices, and edges in the octagonal prism.
 * Solution:** There are 10 faces and 16 vertices. Use Euler’s Theorem, to solve for [[image:http://www.ck12.org/ck12/ucs/?math=E caption="E"]].
 * Example 3:** In a six-faced polyhedron, there are 10 edges. How many vertices does the polyhedron have?
 * Solution:** Solve for [[image:http://www.ck12.org/ck12/ucs/?math=V caption="V"]]in Euler’s Theorem.
 * Example 4:** A three-dimensional figure has 10 vertices, 5 faces, and 12 edges. Is it a polyhedron?
 * Solution:** Plug in all three numbers into Euler’s Theorem.

Regular Polyhedra
All regular polyhedron are **//convex.//** A **//concave//** polyhedron “caves in.” There are only **//five regular polyhedra, called the Platonic solids.//**
 * Regular Polyhedron:** A polyhedron where all the faces are congruent regular polygons.
 * Regular Tetrahedron:** A 4-faced polyhedron and all the faces are equilateral triangles.
 * Cube:** A 6-faced polyhedron and all the faces are squares.
 * Regular Octahedron:** An 8-faced polyhedron and all the faces are equilateral triangles.
 * Regular Dodecahedron:** A 12-faced polyhedron and all the faces are regular pentagons.
 * Regular Icosahedron:** A 20-faced polyhedron and all the faces are equilateral triangles.

Cross-Sections
One way to “view” a three-dimensional figure in a two-dimensional plane, like in this text, is to use cross-sections. The cross-section of the peach plane and the tetrahedron is a //triangle.// a) b) c) a) Square b) Rhombus c) Hexagon
 * Cross-Section:** The intersection of a plane with a solid.
 * Example 5:** What is the shape formed by the intersection of the plane and the regular octahedron?
 * Solution:**

Nets
There are several different nets of any polyhedron. For example, this net could have the triangles anywhere along the top or bottom of the three rectangles. Click the site [] to see a few animations of other nets.
 * Net:** An unfolded, flat representation of the sides of a three-dimensional shape.
 * Example 6:** What kind of figure does this net create?
 * Solution:** The net creates a rectangular prism.
 * Example 7:** Draw a net of the right triangular prism below.
 * Solution:** The net will have two triangles and three rectangles. The rectangles are different sizes and the two triangles are the same.
 * Know What? Revisited** The net of the shape is a regular tetrahedron.

Review Questions
Complete the table using Euler’s Theorem. Determine if the following figures are polyhedra. If so, name the figure and find the number of faces, edges, and vertices. Describe the cross section formed by the intersection of the plane and the solid. Draw the net for the following solids. Determine what shape is formed by the following nets.
 * Questions 1-8 are similar to Examples 2-4.
 * Questions 9-14 are similar to Example 1.
 * Questions 15-17 are similar to Example 5.
 * Questions 18-23 are similar to Example 7.
 * Questions 24-29 are similar to Example 6.
 * Question 30 uses Euler’s Theorem.
 * || **//Name//** || **//Faces//** || **//Edges//** || **//Vertices//** ||
 * 1. || Rectangular Prism || 6 || 12 ||  ||
 * 2. || Octagonal Pyramid ||  || 16 || 9 ||
 * 3. || Regular Icosahedron || 20 ||  || 12 ||
 * 4. || Cube ||  || 12 || 8 ||
 * 5. || Triangular Pyramid || 4 ||  || 4 ||
 * 6. || Octahedron || 8 || 12 ||  ||
 * 7. || Heptagonal Prism ||  || 21 || 14 ||
 * 8. || Triangular Prism || 5 || 9 ||  ||
 * 1) [[image:http://www.ck12.org/ck12/images?id=327982 width="75"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327938 width="65"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=327899 width="65"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=328008 width="85"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=327905 width="95"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=327984 width="80"]]
 * 1) [[image:http://www.ck12.org/ck12/images?id=328035 width="75"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327918 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=327957 width="75"]]
 * 1) [[image:http://www.ck12.org/ck12/images?id=328033 width="65"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=328038 width="80"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=327950 width="65"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=328004 width="80"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=327959 width="100"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=327955 width="65"]]
 * 1) [[image:http://www.ck12.org/ck12/images?id=327902 width="115"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327981 width="115"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=327978 width="115"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=328044 width="115"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=327942 width="115"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=327958 width="115"]]
 * 7) A **//truncated icosahedron//**is a polyhedron with 12 regular pentagonal faces and 20 regular hexagonal faces and 90 edges. This icosahedron closely resembles a soccer ball. How many vertices does it have? Explain your reasoning. [[image:http://www.ck12.org/ck12/images?id=327945 width="100"]]

Review Queue Answers

 * 1) There are 8 vertices and 8 edges in an octagon. [[image:http://www.ck12.org/ck12/images?id=330883 width="150"]]
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=5%5E2%20%3D%2025%20%5C%20cm%5E2 caption="5^2 = 25 cm^2"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=330882 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=330910 width="100"]]

Learning Objectives

 * Find the surface area of a prism and cylinder.

Review Queue

 * 1) Find the area of a rectangle with sides:
 * 2) 6 and 9
 * 3) 11 and 4
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=5%20%5Csqrt%7B2%7D caption="5 sqrt{2}"]]and [[image:http://www.ck12.org/ck12/ucs/?math=8%20%5Csqrt%7B6%7D caption="8 sqrt{6}"]]
 * 5) If the area of a square is [[image:http://www.ck12.org/ck12/ucs/?math=36%20%5C%20units%5E2 caption="36 units^2"]], what are the lengths of the sides?
 * 6) If the area of a square is [[image:http://www.ck12.org/ck12/ucs/?math=45%20%5C%20units%5E2 caption="45 units^2"]], what are the lengths of the sides?
 * Know What?** Your parents decide they want to put a pool in the backyard. The shallow end will be 4 ft. and the deep end will be 8 ft. The pool will be 10 ft. by 25 ft. How much siding do they need to cover the sides and bottom of the pool?



Parts of a Prism
The non-base faces are **//lateral faces.//** The edges between the lateral faces are **//lateral edges.//** This is a **//pentagonal prism.//**
 * Prism:** A 3-dimensional figure with 2 congruent bases, in parallel planes, and the other faces are rectangles.
 * Right Prism:** A prism where all the lateral faces are perpendicular to the bases.
 * Oblique Prism:** A prism that leans to one side and the height is outside the prism.

Surface Area of a Prism
Using the net, we have: Looking at the net, the surface area is:
 * Surface Area:** The sum of the areas of the faces.
 * Lateral Area:** The sum of the areas of the **//lateral//** faces.
 * Example 1:** Find the surface area of the prism below.
 * Solution:** Draw the net of the prism.
 * Surface Area of a Right Prism:** The surface area of a right prism is the sum of the area of the bases and the area of each rectangular lateral face.
 * Example 2:** Find the surface area of the prism below.
 * Solution:** This is a right triangular prism. To find the surface area, we need to find the length of the hypotenuse of the base because it is the width of one of the lateral faces.

Cylinders
A cylinder has a **//radius//** and a **//height.//** A cylinder can also be **//oblique//**, like the one on the far right.
 * Cylinder:** A solid with congruent circular bases that are in parallel planes. The space between the circles is enclosed.

Surface Area of a Right Cylinder
Let’s find the net of a right cylinder. One way to do this is to take the label off of a soup can. The label is a rectangle where the height is the height of the cylinder and the base is the circumference of the circle. To see an animation of the surface area, click [], by Russell Knightley. Now, we can find the surface area. The hypotenuse of the triangle bases is 13,. Let’s fill in what we know.
 * Surface Area of a Right Cylinder:** [[image:http://www.ck12.org/ck12/ucs/?math=SA%3D2%20%5Cpi%20r%5E2%2B2%20%5Cpi%20rh caption="SA=2 pi r^2+2 pi rh"]].
 * Example 3:** Find the surface area of the cylinder.
 * Solution:** [[image:http://www.ck12.org/ck12/ucs/?math=r%20%3D%204 caption="r = 4"]]and [[image:http://www.ck12.org/ck12/ucs/?math=h%20%3D%2012 caption="h = 12"]].
 * Example 4:** The circumference of the base of a cylinder is [[image:http://www.ck12.org/ck12/ucs/?math=16%20%5Cpi caption="16 pi"]]and the height is 21. Find the surface area of the cylinder.
 * Solution:** We need to solve for the radius, using the circumference.
 * Example 5:** **//Algebra Connection//** The total surface area of the triangular prism is [[image:http://www.ck12.org/ck12/ucs/?math=540%20%5C%20units%5E2 caption="540 units^2"]]. What is [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]]?
 * Solution:** The total surface area is equal to:
 * Know What? Revisited** To the right is the net of the pool (minus the top). From this, we can see that your parents would need 670 square feet of siding.

Review Questions
Use the right triangular prism to answer questions 6-9. Find the surface area of the following solids. Round your answer to the nearest hundredth. Use the diagram below for questions 25-30. The barn is shaped like a pentagonal prism with dimensions shown in feet.
 * Questions 1-9 are similar to Examples 1 and 2.
 * Question 10 uses the definition of lateral and total surface area.
 * Questions 11-18 are similar to Examples 1-3.
 * Questions 19-21 are similar to Example 5.
 * Questions 22-24 are similar to Example 4.
 * Questions 25-30 use the Pythagorean Theorem and are similar to Examples 1-3.
 * 1) What type of prism is this? [[image:http://www.ck12.org/ck12/images?id=328025 width="150"]]
 * 2) Draw the net of this prism.
 * 3) Find the area of the bases.
 * 4) Find the area of lateral faces, or the lateral surface area.
 * 5) Find the total surface area of the prism.
 * 1) What shape are the bases of this prism? What are their areas?
 * 2) What are the dimensions of each of the lateral faces? What are their areas?
 * 3) Find the lateral surface area of the prism.
 * 4) Find the total surface area of the prism.
 * 5) **//Writing//** Describe the difference between //lateral// surface area and //total// surface area.
 * 6) Fuzzy dice are cubes with 4 inch sides. [[image:http://www.ck12.org/ck12/images?id=327979 width="125"]]
 * 7) What is the surface area of one die?
 * 8) Typically, the dice are sold in pairs. What is the surface area of two dice?
 * 9) A right cylinder has a 7 cm radius and a height of 18 cm. Find the surface area.
 * 1) bases are isosceles trapezoids [[image:http://www.ck12.org/ck12/images?id=327943 width="125"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327937 width="125"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=327935 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=327921 width="125"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=327941 width="140"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=328048 width="125"]]
 * //Algebra Connection//** Find the value of [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]], given the surface area.
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=SA%20%3D%20432%20%5C%20units%5E2 caption="SA = 432 units^2"]][[image:http://www.ck12.org/ck12/images?id=327914 width="100"]]
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=SA%20%3D%201536%20%5Cpi%20%5C%20units%5E2 caption="SA = 1536 pi units^2"]][[image:http://www.ck12.org/ck12/images?id=328027 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=SA%20%3D%201568%20%5C%20units%5E2 caption="SA = 1568 units^2"]][[image:http://www.ck12.org/ck12/images?id=327901 width="125"]]
 * 4) The area of the base of a cylinder is [[image:http://www.ck12.org/ck12/ucs/?math=25%20%5Cpi%20%5C%20in%5E2 caption="25 pi in^2"]]and the height is 6 in. Find the //lateral// surface area.
 * 5) The circumference of the base of a cylinder is [[image:http://www.ck12.org/ck12/ucs/?math=80%20%5Cpi%20%5C%20cm caption="80 pi cm"]]and the height is 36 cm. Find the total surface area.
 * 6) The lateral surface area of a cylinder is [[image:http://www.ck12.org/ck12/ucs/?math=30%20%5Cpi%20%5C%20m%5E2 caption="30 pi m^2"]]and the height is 5m. What is the radius?
 * 1) What is the width of the roof? (HINT: Use the Pythagorean Theorem)
 * 2) What is the area of the roof? (Both sides)
 * 3) What is the floor area of the barn?
 * 4) What is the area of the rectangular sides of the barn?
 * 5) What is the area of the two pentagon sides of the barn? (HINT: Find the area of two congruent trapezoids for each side)
 * 6) Find the total surface area of the barn (Roof and sides).

Review Queue Answers

 * 1) 54
 * 2) 44
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=80%20%5Csqrt%7B3%7D caption="80 sqrt{3}"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=s%20%3D%206 caption="s = 6"]]
 * 5) [[image:http://www.ck12.org/ck12/ucs/?math=s%20%3D%203%20%5Csqrt%7B5%7D caption="s = 3 sqrt{5}"]]

Learning Objectives

 * Find the surface area of pyramids and cones.

Review Queue

 * 1) A rectangular prism has sides of 5 cm, 6 cm, and 7 cm. What is the surface area?
 * 2) A cylinder has a diameter of 10 in and a height of 25 in. What is the surface area?
 * 3) A cylinder has a circumference of [[image:http://www.ck12.org/ck12/ucs/?math=72%20%5Cpi%20%5C%20ft caption="72 pi ft"]]. and a height of 24 ft. What is the surface area?
 * 4) Draw the net of a square pyramid.
 * Know What?** A typical waffle cone is 6 inches tall and has a diameter of 2 inches. What is the surface area of the waffle cone? (You may assume that the cone is straight across at the top)

Parts of a Pyramid
The edges between the lateral faces are **//lateral edges.//** The edges between the base and the lateral faces are **//base edges.//** All regular pyramids also have a **//slant height//** which is the height of a lateral face. A non-regular pyramid does not have a slant height.
 * Pyramid:** A solid with one **//base//** and the **//lateral faces//** meet at a common **//vertex.//**
 * Regular Pyramid:** A pyramid where the base is a regular polygon.
 * Example 1:** Find the slant height of the square pyramid.
 * Solution:** The slant height is the hypotenuse of the right triangle formed by the height and half the base length. Use the Pythagorean Theorem.

Surface Area of a Regular Pyramid
Using the slant height, which is labeled, the area of each triangular face is. From this example, we see that the formula for a square pyramid is: is the area of the base and is the number of triangles. The net shows the surface area of a pyramid. If you ever forget the formula, use the net. The surface area is:
 * Example 2:** Find the surface area of the pyramid from Example 1.
 * Solution:** The four triangular faces are [[image:http://www.ck12.org/ck12/ucs/?math=4%20%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%20bl%20%5Cright%29%3D2%2816%29%5Cleft%288%5Csqrt%7B10%7D%5Cright%29%3D256%20%5Csqrt%7B10%7D caption="4 left(frac{1}{2} bl right)=2(16)left(8sqrt{10}right)=256 sqrt{10}"]]. To find the total surface area, we also need the area of the base, which is [[image:http://www.ck12.org/ck12/ucs/?math=16%5E2%20%3D%20256 caption="16^2 = 256"]]. The total surface area is [[image:http://www.ck12.org/ck12/ucs/?math=256%20%5Csqrt%7B10%7D%2B256%20%5Capprox%201065.54%20%5C%20units%5E2 caption="256 sqrt{10}+256 approx 1065.54 units^2"]].
 * Surface Area of a Regular Pyramid:** If [[image:http://www.ck12.org/ck12/ucs/?math=B caption="B"]]is the area of the base, then [[image:http://www.ck12.org/ck12/ucs/?math=SA%3DB%2B%5Cfrac%7B1%7D%7B2%7D%20nbl caption="SA=B+frac{1}{2} nbl"]].
 * Example 3:** Find the area of the **//regular//** triangular pyramid.
 * Solution:** “Regular” tells us the base is an equilateral triangle. Let’s draw it and find its area.
 * Example 4:** If the lateral surface area of a square pyramid is [[image:http://www.ck12.org/ck12/ucs/?math=72%20%5C%20ft%5E2 caption="72 ft^2"]]and the base edge is equal to the slant height. What is the length of the base edge?
 * Solution:** In the formula for surface area, the lateral surface area is [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%7B2%7D%20nbl caption="frac{1}{2} nbl"]]. We know that [[image:http://www.ck12.org/ck12/ucs/?math=n%20%3D%204 caption="n = 4"]]and [[image:http://www.ck12.org/ck12/ucs/?math=b%20%3D%20l caption="b = l"]]. Let’s solve for [[image:http://www.ck12.org/ck12/ucs/?math=b caption="b"]].

Surface Area of a Cone
A cone has a slant height, just like a pyramid. A cone is generated from rotating a right triangle, around one leg, in a circle. Area of the base: Area of the sides: The surface area would be.
 * Cone:** A solid with a circular base and sides taper up towards a vertex.
 * Surface Area of a Right Cone:** [[image:http://www.ck12.org/ck12/ucs/?math=SA%3D%5Cpi%20r%5E2%2B%5Cpi%20rl caption="SA=pi r^2+pi rl"]].
 * Example 5:** What is the surface area of the cone?
 * Solution:** First, we need to find the slant height. Use the Pythagorean Theorem.
 * Example 6:** The surface area of a cone is [[image:http://www.ck12.org/ck12/ucs/?math=36%20%5Cpi caption="36 pi"]]and the radius is 4 units. What is the slant height?
 * Solution:** Plug in what you know into the formula for the surface area of a cone and solve for [[image:http://www.ck12.org/ck12/ucs/?math=l caption="l"]].
 * Know What? Revisited** The standard cone has a surface area of [[image:http://www.ck12.org/ck12/ucs/?math=%5Cpi%2B%20%5Csqrt%7B35%7D%5Cpi%20%5Capprox%2021.73%20%5C%20in%5E2 caption="pi+ sqrt{35}pi approx 21.73 in^2"]].

Review Questions
Fill in the blanks about the diagram to the left. Use the cone to fill in the blanks. For questions 11-13, sketch each of the following solids and answer the question. Your drawings should be to scale, but not one-to-one. Leave your answer in simplest radical form. Find the slant height,, of one lateral face in each pyramid or of the cone. Round your answer to the nearest hundredth. Find the area of a lateral face of the regular pyramid. Round your answers to the nearest hundredth. Find the surface area of the regular pyramids and right cones. Round your answers to 2 decimal places. The traffic cone is cut off at the top and the base is a square with 24 in sides. Round answers to the nearest hundredth.
 * Questions 1-10 use the definitions of pyramids and cones.
 * Questions 11-19 are similar to Example 1.
 * Questions 20-26 are similar to Examples 2, 3, and 5.
 * Questions 27-31 are similar to Examples 4 and 6.
 * Questions 32-25 are similar to Example 5.
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]]is the ___.__
 * 2) The slant height is __.__
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=y caption="y"]]is the_.
 * 4) The height is.
 * 5) The base is ___.__
 * 6) The base edge is__.
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=v caption="v"]]is the ___.__
 * 2) The height of the cone is.
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]]is a __and it is the__ _ of the cone.
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=w caption="w"]]is the _.
 * 1) Draw a right cone with a radius of 5 cm and a height of 15 cm. What is the slant height?
 * 2) Draw a square pyramid with an edge length of 9 in and a 12 in height. Find the slant height.
 * 3) Draw an equilateral triangle pyramid with an edge length of 6 cm and a height of 6 cm. What is the height of the base?
 * 1) [[image:http://www.ck12.org/ck12/images?id=327934 width="80"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327906 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=327911 width="100"]]
 * 1) [[image:http://www.ck12.org/ck12/images?id=328050 width="80"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327917 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=327972 width="100"]]
 * 1) [[image:http://www.ck12.org/ck12/images?id=328039 width="80"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327929 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=327975 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=327891 width="80"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=327946 width="100"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=328036 width="100"]]
 * 7) A **//regular tetrahedron//**has four equilateral triangles as its faces.
 * 8) Find the height of one of the faces if the edge length is 6 units.
 * 9) Find the area of one face.
 * 10) Find the total surface area of the regular tetrahedron.
 * 11) If the lateral surface area of a cone is [[image:http://www.ck12.org/ck12/ucs/?math=30%20%5Cpi%20%5C%20cm%5E2 caption="30 pi cm^2"]]and the radius is 5 cm, what is the slant height?
 * 12) If the surface area of a cone is [[image:http://www.ck12.org/ck12/ucs/?math=105%20%5Cpi%20%5C%20cm%5E2 caption="105 pi cm^2"]]and the slant height is 8 cm, what is the radius?
 * 13) If the surface area of a square pyramid is [[image:http://www.ck12.org/ck12/ucs/?math=40%20%5C%20ft%5E2 caption="40 ft^2"]]and the base edge is 4 ft, what is the slant height?
 * 14) If the lateral area of a square pyramid is [[image:http://www.ck12.org/ck12/ucs/?math=800%20%5C%20in%5E2 caption="800 in^2"]]and the slant height is 16 in, what is the length of the base edge?
 * 15) If the lateral area of a regular triangle pyramid is [[image:http://www.ck12.org/ck12/ucs/?math=252%20%5C%20in%5E2 caption="252 in^2"]]and the base edge is 8 in, what is the slant height?
 * 1) Find the area of the entire square. Then, subtract the area of the base of the cone.
 * 2) Find the lateral area of the cone portion (include the 4 inch cut off top of the cone).
 * 3) Subtract the cut-off top of the cone, to only have the lateral area of the cone portion of the traffic cone.
 * 4) Combine your answers from #27 and #30 to find the entire surface area of the traffic cone.

Review Queue Answers

 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=2%285%20%5Ccdot%206%29%20%2B%202%285%20%5Ccdot%207%29%20%2B%202%286%20%5Ccdot%207%29%20%3D%20214%20%5C%20cm%5E2 caption="2(5 cdot 6) + 2(5 cdot 7) + 2(6 cdot 7) = 214 cm^2"]]
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=2%2815%20%5Ccdot%2018%29%20%2B%202%2815%20%5Ccdot%2021%29%20%2B%202%2818%20%5Ccdot%2021%29%20%3D%201926%20%5C%20cm%5E2 caption="2(15 cdot 18) + 2(15 cdot 21) + 2(18 cdot 21) = 1926 cm^2"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=2%20%5Ccdot%2025%20%5Cpi%20%2B%20250%20%5Cpi%20%3D%20300%20%5Cpi%20%5C%20in%5E2 caption="2 cdot 25 pi + 250 pi = 300 pi in^2"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=36%5E2%20%282%20%5Cpi%29%20%2B%2072%20%5Cpi%20%2824%29%20%3D%204320%20%5Cpi%20%5C%20ft%5E2 caption="36^2 (2 pi) + 72 pi (24) = 4320 pi ft^2"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=330886 width="120"]]

Learning Objectives

 * Find the volume of prisms and cylinders.

Review Queue

 * 1) Define volume in your own words.
 * 2) What is the surface area of a cube with 3 inch sides?
 * 3) A regular octahedron has 8 congruent equilateral triangles as the faces.
 * 4) If each edge is 4 cm, what is the slant height for one face?
 * 5) What is the surface area of one face?
 * 6) What is the total surface area? [[image:http://www.ck12.org/ck12/images?id=328012 width="85"]]
 * Know What?** Let’s fill the pool it with water. The shallow end is 4 ft. and the deep end is 8 ft. The pool is 10 ft. wide by 25 ft. long. How many cubic feet of water is needed to fill the pool?

Volume of a Rectangular Prism
Another way to define volume would be how much a three-dimensional figure can hold. The basic unit of volume is the cubic unit: cubic centimeter, cubic inch , cubic meter , cubic foot. What this postulate tells us is that every solid can be broken down into cubes. For example, if we wanted to find the volume of a cube with 9 inch sides, it would be. These prisms are congruent, so their volumes are congruent. Each layer in Example 1 is the same as the area of the base and the number of layers is the same as the height. This is the formula for volume.
 * Volume:** The measure of how much space a three-dimensional figure occupies.
 * Volume of a Cube Postulate:** [[image:http://www.ck12.org/ck12/ucs/?math=V%3Ds%5E3 caption="V=s^3"]].
 * Volume Congruence Postulate:** If two solids are congruent, then their volumes are congruent.
 * Example 1:** Find the volume of the right rectangular prism below.
 * Solution:** Count the cubes. The bottom layer has 20 cubes, or [[image:http://www.ck12.org/ck12/ucs/?math=4%20%5Ctimes%205 caption="4 times 5"]], and there are 3 layers. There are 60 cubes. The volume is also [[image:http://www.ck12.org/ck12/ucs/?math=60%20%5C%20units%5E3 caption="60 units^3"]].
 * Volume of a Rectangular Prism:** [[image:http://www.ck12.org/ck12/ucs/?math=V%3Dl%20%5Ccdot%20w%20%5Ccdot%20h caption="V=l cdot w cdot h"]].
 * Example 2:** A typical shoe box is 8 in by 14 in by 6 in. What is the volume of the box?
 * Solution:** We can assume that a shoe box is a rectangular prism.

Volume of any Prism
Notcie that is equal to the area of the base of the prism, which we will re-label. “” is not always going to be the same. So, to find the volume of a prism, you would first find the area of the base and then multiply it by the height. Even though the height in this problem does not look like a “height,” it is, according to the formula. Usually, the height of a prism is going to be the last length you need to use.
 * Volume of a Prism:** [[image:http://www.ck12.org/ck12/ucs/?math=V%3DB%20%5Ccdot%20h caption="V=B cdot h"]].
 * Example 3:** You have a small, triangular prism shaped tent. How much volume does it have, once it is set up?
 * Solution:** First, we need to find the area of the base.

Oblique Prisms
Recall that oblique prisms are prisms that lean to one side and the height is outside the prism. What would be the volume of an oblique prism? Consider to piles of books below. Both piles have 15 books, which means they will have the same volume. Cavalieri’s Principle says that leaning does not matter, the volumes are the same. If an oblique prism and a right prism have the same base area and height, then they will have the same volume.
 * Cavalieri’s Principle:** If two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.
 * Example 4:** Find the area of the oblique prism below.
 * Solution:** This is an oblique right trapezoidal prism. Find the area of the trapezoid.

Volume of a Cylinder
If we use the formula for the volume of a prism,, we can find the volume of a cylinder. In the case of a cylinder, the base is the area of a circle. Like a prism, Cavalieri’s Principle holds. The total volume is.
 * Volume of a Cylinder:** [[image:http://www.ck12.org/ck12/ucs/?math=V%3D%5Cpi%20r%5E2%20h caption="V=pi r^2 h"]].
 * Example 5:** Find the volume of the cylinder.
 * Solution:** If the diameter is 16, then the radius is 8.
 * Example 6:** Find the volume of the cylinder.
 * Solution:** [[image:http://www.ck12.org/ck12/ucs/?math=V%3D%5Cpi%206%5E2%20%2815%29%3D540%20%5Cpi%20%5C%20units%5E3 caption="V=pi 6^2 (15)=540 pi units^3"]]
 * Example 7:** If the volume of a cylinder is [[image:http://www.ck12.org/ck12/ucs/?math=484%20%5Cpi%20%5C%20in%5E3 caption="484 pi in^3"]]and the height is 4 in, what is the radius?
 * Solution:** Solve for [[image:http://www.ck12.org/ck12/ucs/?math=r caption="r"]].
 * Example 8:** Find the volume of the solid below.
 * Solution:** This solid is a parallelogram-based prism with a cylinder cut out of the middle.
 * Know What? Revisited** Even though it doesn’t look like it, the trapezoid is the base of this prism. The area of the trapezoids are [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%7B2%7D%20%284%2B8%2925%3D150%20%5C%20ft%5E2 caption="frac{1}{2} (4+8)25=150 ft^2"]]. [[image:http://www.ck12.org/ck12/ucs/?math=V%3D150%2810%29%3D1500%20%5C%20ft%5E3 caption="V=150(10)=1500 ft^3"]]

Review Questions
Use the right triangular prism to answer questions 9 and 10. Find the volume of the following solids. Round your answers to the nearest hundredth. The bases of the prism are squares and a cylinder is cut out of the center. This is a prism with half a cylinder on the top.
 * Question 1 uses the volume formula for a cylinder.
 * Questions 2-4 are similar to Example 1.
 * Questions 5-18 are similar to Examples 2-6.
 * Questions 19-24 are similar to Example 7.
 * Questions 25-30 are similar to Example 8.
 * 1) Two cylinders have the same surface area. Do they have the same volume? How do you know?
 * 2) How many one-inch cubes can fit into a box that is 8 inches wide, 10 inches long, and 12 inches tall? Is this the same as the volume of the box?
 * 3) A cereal box in 2 inches wide, 10 inches long and 14 inches tall. How much cereal does the box hold?
 * 4) A can of soda is 4 inches tall and has a diameter of 2 inches. How much soda does the can hold? Round your answer to the nearest hundredth.
 * 5) A cube holds [[image:http://www.ck12.org/ck12/ucs/?math=216%20%5C%20in%5E3 caption="216 in^3"]]. What is the length of each edge?
 * 6) A cube has sides that are 8 inches. What is the volume?
 * 7) A cylinder has [[image:http://www.ck12.org/ck12/ucs/?math=r%20%3D%20h caption="r = h"]]and the radius is 4 cm. What is the volume?
 * 8) A cylinder has a volume of [[image:http://www.ck12.org/ck12/ucs/?math=486%20%5Cpi%20%5C%20ft.%5E3 caption="486 pi ft.^3"]]. If the height is 6 ft., what is the diameter?
 * 1) What is the length of the third base edge?
 * 2) Find the volume of the prism.
 * 3) Fuzzy dice are cubes with 4 inch sides. [[image:http://www.ck12.org/ck12/images?id=327979 width="110"]]
 * 4) What is the volume of one die?
 * 5) What is the volume of both dice?
 * 6) A right cylinder has a 7 cm radius and a height of 18 cm. Find the volume.
 * 1) [[image:http://www.ck12.org/ck12/images?id=327943 width="115"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327937 width="115"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=327935 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=327921 width="115"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=327941 width="115"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=328048 width="115"]]
 * //Algebra Connection//** Find the value of [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]], given the surface area.
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=V%3D504%20%5C%20units%5E3 caption="V=504 units^3"]][[image:http://www.ck12.org/ck12/images?id=327914 width="85"]]
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=V%3D6144%20%5Cpi%20%5C%20units%5E3 caption="V=6144 pi units^3"]][[image:http://www.ck12.org/ck12/images?id=328027 width="85"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=V%3D2688%20%5C%20units%5E3 caption="V=2688 units^3"]][[image:http://www.ck12.org/ck12/images?id=327901 width="115"]]
 * 4) The area of the base of a cylinder is [[image:http://www.ck12.org/ck12/ucs/?math=49%20%5Cpi%20%5C%20in%5E2 caption="49 pi in^2"]]and the height is 6 in. Find the volume.
 * 5) The circumference of the base of a cylinder is [[image:http://www.ck12.org/ck12/ucs/?math=80%20%5Cpi%20%5C%20cm caption="80 pi cm"]]and the height is 15 cm. Find the volume.
 * 6) The lateral surface area of a cylinder is [[image:http://www.ck12.org/ck12/ucs/?math=30%20%5Cpi%20%5C%20m%5E2 caption="30 pi m^2"]]and the circumference is [[image:http://www.ck12.org/ck12/ucs/?math=10%20%5Cpi%20%5C%20m caption="10 pi m"]]. What is the volume of the cylinder?
 * 1) Find the volume of the prism.
 * 2) Find the volume of the cylinder in the center.
 * 3) Find the volume of the figure.
 * 1) Find the volume of the prism.
 * 2) Find the volume of the half-cylinder.
 * 3) Find the volume of the entire figure.

Review Queue Answers

 * 1) The amount a three-dimensional figure can hold.
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=54%20%5C%20in%5E2 caption="54 in^2"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=2%20%5Csqrt%7B3%7D caption="2 sqrt{3}"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%7B2%7D%20%5Ccdot%204%20%5Ccdot%202%20%5Csqrt%7B3%7D%20%3D%204%20%5Csqrt%7B3%7D caption="frac{1}{2} cdot 4 cdot 2 sqrt{3} = 4 sqrt{3}"]]
 * 5) [[image:http://www.ck12.org/ck12/ucs/?math=8%20%5Ccdot%204%20%5Csqrt%7B3%7D%20%3D%2032%20%5Csqrt%7B3%7D caption="8 cdot 4 sqrt{3} = 32 sqrt{3}"]]

Learning Objectives

 * Find the volume of pyramids and cones.

Review Queue

 * 1) Find the volume of a square prism with 8 inch base edges and a 12 inch height.
 * 2) Find the volume of a cylinder with a //__diameter__// of 8 inches and a height of 12 inches.
 * 3) Find the surface area of a square pyramid with 10 inch base edges and a height of 12 inches.
 * Know What?** The Khafre Pyramid is a pyramid in Giza, Egypt. It is a square pyramid with a base edge of 706 feet and an original height of 407.5 feet. What was the original volume of the Khafre Pyramid?

Volume of a Pyramid
The volume of a pyramid is closely related to the volume of a prism with the same sized base. Tools needed: pencil, paper, scissors, tape, ruler, dry rice.
 * Investigation 11-1: Finding the Volume of a Pyramid**
 * 1) Make an open net (omit one base) of a cube, with 2 inch sides. [[image:http://www.ck12.org/ck12/images?id=327927 width="100"]]
 * 2) Cut out the net and tape up the sides to form an open cube. [[image:http://www.ck12.org/ck12/images?id=327925 width="85"]]
 * 3) Make an open net (no base) of a square pyramid, with lateral edges of 2.5 inches and base edges of 2 inches. [[image:http://www.ck12.org/ck12/images?id=327965 width="75"]]
 * 4) Cut out the net and tape up the sides to form an open pyramid. [[image:http://www.ck12.org/ck12/images?id=327991 width="85"]]
 * 5) Fill the pyramid with dry rice and dump the rice into the open cube. Repeat this until you have filled the cube?
 * Volume of a Pyramid:** [[image:http://www.ck12.org/ck12/ucs/?math=V%3D%5Cfrac%7B1%7D%7B3%7D%20Bh caption="V=frac{1}{3} Bh"]].
 * Example 1:** Find the volume of the pyramid.
 * Solution:** [[image:http://www.ck12.org/ck12/ucs/?math=V%3D%5Cfrac%7B1%7D%7B3%7D%20%2812%5E2%20%2912%3D576%20%5C%20units%5E3 caption="V=frac{1}{3} (12^2 )12=576 units^3"]]
 * Example 2a:** Find the height of the pyramid.
 * Solution:** In this example, we are given the //slant// height. Use the Pythagorean Theorem.
 * Example 2b:** Find the volume of the pyramid in Example 2a.
 * Solution:** [[image:http://www.ck12.org/ck12/ucs/?math=V%3D%5Cfrac%7B1%7D%7B3%7D%20%2814%5E2%29%2824%29%3D1568%20%5C%20units%5E3 caption="V=frac{1}{3} (14^2)(24)=1568 units^3"]].
 * Example 3:** Find the volume of the pyramid.
 * Solution:** The base of the pyramid is a right triangle. The area of the base is [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%7B2%7D%20%2814%29%288%29%3D56%20%5C%20units%5E2 caption="frac{1}{2} (14)(8)=56 units^2"]].
 * Example 4:** A rectangular pyramid has a base area of [[image:http://www.ck12.org/ck12/ucs/?math=56%20%5C%20cm%5E2 caption="56 cm^2"]]and a volume of [[image:http://www.ck12.org/ck12/ucs/?math=224%20%5C%20cm%5E3 caption="224 cm^3"]]. What is the height of the pyramid?
 * Solution:**

Volume of a Cone
This is the the same relationship as a pyramid’s volume with a prism’s volume.
 * Volume of a Cone:** [[image:http://www.ck12.org/ck12/ucs/?math=V%3D%5Cfrac%7B1%7D%7B3%7D%20%5Cpi%20r%5E2%20h caption="V=frac{1}{3} pi r^2 h"]].
 * Example 5:** Find the volume of the cone.
 * Solution:** First, we need the height. Use the Pythagorean Theorem.
 * Example 6:** Find the volume of the cone.
 * Solution:** We can use the same volume formula. Find the //radius.//
 * Example 7:** The volume of a cone is [[image:http://www.ck12.org/ck12/ucs/?math=484%20%5Cpi%20%5C%20cm%5E3 caption="484 pi cm^3"]]and the height is 12 cm. What is the radius?
 * Solution:** Plug in what you know to the volume formula.

Composite Solids
The total volume is.
 * Example 8:** Find the volume of the composite solid. All bases are squares.
 * Solution:** This is a square prism with a square pyramid on top. First, we need the height of the pyramid portion. Using the Pythagorean Theorem, we have, [[image:http://www.ck12.org/ck12/ucs/?math=h%3D%5Csqrt%7B25%5E2-24%5E2%7D%3D7 caption="h=sqrt{25^2-24^2}=7"]].
 * Know What? Revisited** The original volume of the pyramid is [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%7B3%7D%20%28706%5E2%29%28407.5%29%20%0A%5Capprox%2067%2C704%2C223.33%20%5C%20ft%5E3 caption="frac{1}{3} (706^2)(407.5) approx 67,704,223.33 ft^3"]].

Review Questions
Find the volume of each regular pyramid and right cone. Round any decimal answers to the nearest hundredth. The bases of these pyramids are either squares or equilateral triangles. Find the volume of the following non-regular pyramids and cones. Round any decimal answers to the nearest hundredth. A **//regular tetrahedron//** has four equilateral triangles as its faces. Use the diagram to answer questions 14-16. Round your answers to the nearest hundredth. A **//regular octahedron//** has eight equilateral triangles as its faces. Use the diagram to answer questions 17-21. Round your answers to the nearest hundredth. For questions 23-31, round your answer to the nearest hundredth. The solid to the right is a cube with a cone cut out. The solid to the left is a cylinder with a cone on top.
 * Questions 1-13 are similar to Examples 1-3, 5 and 6.
 * Questions 14-22 are similar to Examples 4 and 7.
 * Questions 23-31 are similar to Example 8.
 * 1) [[image:http://www.ck12.org/ck12/images?id=328050 width="85"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327917 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=328039 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=327929 width="100"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=327891 width="85"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=327946 width="100"]]
 * 7) [[image:http://www.ck12.org/ck12/images?id=328036 width="100"]]
 * 1) [[image:http://www.ck12.org/ck12/images?id=328046 width="100"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327962 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=327967 width="125"]]
 * 4) base is a rectangle [[image:http://www.ck12.org/ck12/images?id=327956 width="125"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=328010 width="85"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=328002 width="100"]]
 * 1) What is the area of the base of this regular tetrahedron?
 * 2) What is the height of this figure? Be careful!
 * 3) Find the volume.
 * 1) //Describe// how you would find the volume of this figure.
 * 2) Find the volume.
 * 3) The volume of a square pyramid is 72 square inches and the base edge is 4 inches. What is the height?
 * 4) If the volume of a cone is [[image:http://www.ck12.org/ck12/ucs/?math=30%20%5Cpi%20%5C%20cm%5E3 caption="30 pi cm^3"]]and the radius is 5 cm, what is the height?
 * 5) If the volume of a cone is [[image:http://www.ck12.org/ck12/ucs/?math=105%20%5Cpi%20%5C%20cm%5E3 caption="105 pi cm^3"]]and the height is 35 cm, what is the radius?
 * 6) The volume of a triangle pyramid is [[image:http://www.ck12.org/ck12/ucs/?math=170%20%5C%20in%5E3 caption="170 in^3"]]and the base area is [[image:http://www.ck12.org/ck12/ucs/?math=34%20%5C%20in%5E2 caption="34 in^2"]]. What is the height of the pyramid?
 * 1) Find the volume of the base prism. [[image:http://www.ck12.org/ck12/images?id=328042 width="125"]]
 * 2) Find the volume of the pyramid.
 * 3) Find the volume of the entire solid.
 * 1) Find the volume of the cube.
 * 2) Find the volume of the cone.
 * 3) Find the volume of the entire solid.
 * 1) Find the volume of the cylinder.
 * 2) Find the volume of the cone.
 * 3) Find the volume of the entire solid.

Review Queue Answers

 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=%288%5E2%29%2812%29%20%3D%20768%20%5C%20in%5E3 caption="(8^2)(12) = 768 in^3"]]
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=%284%5E2%29%2812%29%20%5Cpi%20%3D%20192%20%5Cpi%20%5Capprox%20603.19 caption="(4^2)(12) pi = 192 pi approx 603.19"]]
 * 3) Find slant height, [[image:http://www.ck12.org/ck12/ucs/?math=l%20%3D%2013 caption="l = 13"]]. [[image:http://www.ck12.org/ck12/ucs/?math=SA%20%3D%20100%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20%2840%29%2813%29%3D360%20%5C%20in%5E2 caption="SA = 100 + frac{1}{2} (40)(13)=360 in^2"]]

Learning Objectives

 * Find the surface area of a sphere.
 * Find the volume of a sphere.

Review Queue

 * 1) List three spheres you would see in real life.
 * 2) Find the area of a circle with a 6 cm radius.
 * 3) Find the volume of a cylinder with the circle from #2 as the base and a height of 5 cm.
 * Know What?** A regulation bowling ball is a sphere with a circumference of 27 inches. Find the radius of a bowling ball, its surface area and volume. You may assume the bowling ball does not have any finger holes. Round your answers to the nearest hundredth.

Defining a Sphere
A sphere is the last of the three-dimensional shapes that we will find the surface area and volume of. Think of a sphere as a three-dimensional circle. The **//radius//** has an endpoint on the sphere and the other endpoint is the center. The **//diameter//** must contain the center. A great circle is the largest circle cross section in a sphere. **//The circumference of a sphere is the circumference of a great circle//**. Every great circle divides a sphere into two congruent **//hemispheres.//**
 * Sphere:** The set of all points, in three-dimensional space, which are equidistant from a point.
 * Great Circle:** A cross section of a sphere that contains the diameter.
 * Example 1:** The circumference of a sphere is [[image:http://www.ck12.org/ck12/ucs/?math=26%20%5Cpi%20%5C%20feet caption="26 pi feet"]]. What is the radius of the sphere?
 * Solution:** The circumference is referring to the circumference of a great circle. Use [[image:http://www.ck12.org/ck12/ucs/?math=C%20%3D%202%20%5Cpi%20r caption="C = 2 pi r"]].

Surface Area of a Sphere
The best way to understand the surface area of a sphere is to watch the link by Russell Knightley, []
 * Surface Area of a Sphere:** [[image:http://www.ck12.org/ck12/ucs/?math=SA%3D4%20%5Cpi%20r%5E2 caption="SA=4 pi r^2"]].
 * Example 2:** Find the surface area of a sphere with a radius of 14 feet.
 * Solution:**
 * Example 3:** Find the surface area of the figure below.
 * Solution:** Be careful when finding the surface area of a hemisphere because you need to include the area of the base.
 * Example 4:** The surface area of a sphere is [[image:http://www.ck12.org/ck12/ucs/?math=100%20%5Cpi%20%5C%20in%5E2 caption="100 pi in^2"]]. What is the radius?
 * Solution:**
 * Example 5:** Find the surface area of the following solid.
 * Solution:** This solid is a cylinder with a hemisphere on top. It is one solid, so do not include the bottom of the hemisphere or the top of the cylinder.

Volume of a Sphere
To see an animation of the volume of a sphere, see [] by Russell Knightley. At this point, you will need to take the **//cubed root//** of 3375. Ask your teacher how to do this on your calculator.
 * Volume of a Sphere:** [[image:http://www.ck12.org/ck12/ucs/?math=V%3D%5Cfrac%7B4%7D%7B3%7D%20%5Cpi%20r%5E3 caption="V=frac{4}{3} pi r^3"]].
 * Example 6:** Find the volume of a sphere with a radius of 9 m.
 * Solution:**
 * Example 7:** A sphere has a volume of [[image:http://www.ck12.org/ck12/ucs/?math=14137.167%20%5C%20ft%5E3 caption="14137.167 ft^3"]], what is the radius?
 * Solution:**
 * Example 8:** Find the volume of the following solid.
 * Solution:**
 * Know What? Revisited** The radius would be [[image:http://www.ck12.org/ck12/ucs/?math=27%3D2%20%5Cpi%20r caption="27=2 pi r"]], or [[image:http://www.ck12.org/ck12/ucs/?math=r%3D4.30%20%5C%20inches caption="r=4.30 inches"]]. The surface area would be [[image:http://www.ck12.org/ck12/ucs/?math=4%20%5Cpi%204.3%5E2%20%5Capprox%20232.35%20%5C%20in%5E2 caption="4 pi 4.3^2 approx 232.35 in^2"]], and the volume would be [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B4%7D%7B3%7D%20%5Cpi%204.3%5E3%20%5Capprox%20333.04%20%5C%20in%5E3 caption="frac{4}{3} pi 4.3^3 approx 333.04 in^3"]].

Review Questions
Find the surface area __and__ volume of a sphere with: (Leave your answer in terms of ) Find the surface area of the following shapes. Leave your answers in terms of. Find the volume of the following shapes. Round your answers to the nearest hundredth. Tennis balls with a 3 inch diameter are sold in cans of three. The can is a cylinder. Round your answers to the nearest hundredth.
 * Questions 1-3 look at the definition of a sphere.
 * Questions 4-17 are similar to Examples 1, 2, 4, 6 and 7.
 * Questions 18-21 are similar to Example 3 and 5.
 * Questions 22-25 are similar to Example 8.
 * Question 26 is a challenge.
 * Questions 27-29 are similar to Example 8.
 * Question 30 analyzes the formula for the surface area of a sphere.
 * 1) Are there any cross-sections of a sphere that are not a circle? Explain your answer.
 * 2) List all the parts of a sphere that are the **//same//** as a circle.
 * 3) List any parts of a sphere that a circle does not have.
 * 1) a radius of 8 in.
 * 2) a diameter of 18 cm.
 * 3) a radius of 20 ft.
 * 4) a diameter of 4 m.
 * 5) a radius of 15 ft.
 * 6) a diameter of 32 in.
 * 7) a circumference of [[image:http://www.ck12.org/ck12/ucs/?math=26%20%5Cpi%20%5C%20cm caption="26 pi cm"]].
 * 8) a circumference of [[image:http://www.ck12.org/ck12/ucs/?math=50%20%5Cpi%20%5C%20yds caption="50 pi yds"]].
 * 9) The surface area of a sphere is [[image:http://www.ck12.org/ck12/ucs/?math=121%20%5Cpi%20%5C%20in%5E2 caption="121 pi in^2"]]. What is the radius?
 * 10) The volume of a sphere is [[image:http://www.ck12.org/ck12/ucs/?math=47916%20%5Cpi%20%5C%20m%5E3 caption="47916 pi m^3"]]. What is the radius?
 * 11) The surface area of a sphere is [[image:http://www.ck12.org/ck12/ucs/?math=4%20%5Cpi%20%5C%20ft%5E2 caption="4 pi ft^2"]]. What is the volume?
 * 12) The volume of a sphere is [[image:http://www.ck12.org/ck12/ucs/?math=36%20%5Cpi%20%5C%20mi%5E3 caption="36 pi mi^3"]]. What is the surface area?
 * 13) Find the radius of the sphere that has a volume of [[image:http://www.ck12.org/ck12/ucs/?math=335%20%5C%20cm%5E3 caption="335 cm^3"]]. Round your answer to the nearest hundredth.
 * 14) Find the radius of the sphere that has a surface area [[image:http://www.ck12.org/ck12/ucs/?math=225%20%5Cpi%20%5C%20ft%5E2 caption="225 pi ft^2"]].
 * 1) [[image:http://www.ck12.org/ck12/images?id=327992 width="100"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327999 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=327968 width="135"]]
 * 4) You may assume the bottom is //open//. [[image:http://www.ck12.org/ck12/images?id=327908 width="110"]]
 * 1) [[image:http://www.ck12.org/ck12/images?id=327992 width="100"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327999 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=327968 width="135"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=327908 width="110"]]
 * 5) A sphere has a radius of 5 cm. A right cylinder has the same radius and volume. Find the height of the cylinder.
 * 1) What is the volume of one tennis ball?
 * 2) What is the volume of the cylinder?
 * 3) Assume the balls touch the can on the sides, top and bottom. What is the volume of the space //not// occupied by the tennis balls?
 * 4) How does the formula of the surface area of a sphere relate to the area of a circle?

Review Queue Answers

 * 1) //Answers will vary.// Possibilities are any type of ball, certain lights, or the 76/Unical orb.
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=36%20%5Cpi caption="36 pi"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=180%20%5Cpi caption="180 pi"]]

Learning Objectives

 * Find the relationship between similar solids and their surface areas and volumes.

Similar Solids
Recall that two shapes are similar if all the corresponding angles are congruent and the corresponding sides are proportional. The congruent ratios tell us the two prisms are similar. however,. These triangle pyramids are not similar.
 * Similar Solids:** Two solids are similar if they are the same type of solid and their corresponding radii, heights, base lengths, widths, etc. are proportional.
 * Example 1:** Are the two rectangular prisms similar? How do you know?
 * Solution:** Match up the corresponding heights, widths, and lengths.
 * Example 2:** Determine if the two triangular pyramids similar.
 * Solution:** Just like Example 1, let’s match up the corresponding parts.

Surface Areas of Similar Solids
For example, the two rectangles are similar because their sides are in a ratio of 5:8. The area of the larger rectangle is. The area of the smaller rectangle is. Comparing the areas in a ratio, it is. So, what happens with the surface areas of two similar solids? Now, find the ratio of the areas. . The sides are in a ratio of, so the surface areas are in a ratio of.
 * //If two shapes are similar, then the ratio of the area is a square of the scale factor.//**
 * Example 3:** Find the surface area of the two similar rectangular prisms.
 * Solution:**
 * Surface Area Ratio:** If two solids are similar with a scale factor of [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Ba%7D%7Bb%7D caption="frac{a}{b}"]], then the surface areas are in a ratio of [[image:http://www.ck12.org/ck12/ucs/?math=%5Cleft%28%20%5Cfrac%7Ba%7D%7Bb%7D%20%5Cright%29%5E2 caption="left( frac{a}{b} right)^2"]].
 * Example 4:** Two similar cylinders are below. If the ratio of the areas is 16:25, what is the height of the taller cylinder?
 * Solution:** First, we need to take the square root of the area ratio to find the scale factor, [[image:http://www.ck12.org/ck12/ucs/?math=%5Csqrt%7B%5Cfrac%7B16%7D%7B25%7D%7D%3D%5Cfrac%7B4%7D%7B5%7D caption="sqrt{frac{16}{25}}=frac{4}{5}"]]. Set up a proportion to find [[image:http://www.ck12.org/ck12/ucs/?math=h caption="h"]].
 * Example 5:** Using the cylinders from Example 4, if the area of the smaller cylinder is [[image:http://www.ck12.org/ck12/ucs/?math=1536%20%5Cpi%20%5C%20cm%5E2 caption="1536 pi cm^2"]], what is the area of the larger cylinder?
 * Solution:** Set up a proportion using the ratio of the areas, 16:25.

Volumes of Similar Solids
Let’s look at what we know about similar solids so far. If the ratio of the volumes follows the pattern from above, it should be the **//cube//** of the scale factor. The ratio is, which reduces to.
 * || **//Ratios//** || **//Units//** ||
 * **//Scale Factor//** || [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Ba%7D%7Bb%7D caption="frac{a}{b}"]] || in, ft, cm, m, etc. ||
 * **//Ratio of the Surface Areas//** || [[image:http://www.ck12.org/ck12/ucs/?math=%5Cleft%28%5Cfrac%7Ba%7D%7Bb%7D%5Cright%29%5E2 caption="left(frac{a}{b}right)^2"]] || [[image:http://www.ck12.org/ck12/ucs/?math=in%5E2%2C%20ft%5E2%2C%20cm%5E2%2C%20m%5E2 caption="in^2, ft^2, cm^2, m^2"]], etc. ||
 * **//Ratio of the Volumes//** || **??** || [[image:http://www.ck12.org/ck12/ucs/?math=in%5E3%2C%20ft%5E3%2C%20cm%5E3%2C%20m%5E3 caption="in^3, ft^3, cm^3, m^3"]], etc. ||
 * Example 6:** Find the volume of the following rectangular prisms. Then, find the ratio of the volumes.
 * Solution:**
 * Volume Ratio:** If two solids are similar with a scale factor of [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Ba%7D%7Bb%7D caption="frac{a}{b}"]], then the volumes are in a ratio of [[image:http://www.ck12.org/ck12/ucs/?math=%5Cleft%28%20%5Cfrac%7Ba%7D%7Bb%7D%20%5Cright%29%5E3 caption="left( frac{a}{b} right)^3"]].
 * Example 7:** Two spheres have radii in a ratio of 3:4. What is the ratio of their volumes?
 * Solution:** If we cube 3 and 4, we will have the ratio of the volumes. [[image:http://www.ck12.org/ck12/ucs/?math=3%5E3%3A4%5E3%20%3D%2027%3A64 caption="3^3:4^3 = 27:64"]].
 * Example 8:** If the ratio of the volumes of two similar prisms is 125:8, what is the scale factor?
 * Solution:** Take the **//cubed root//** of 125 and 8 to find the scale factor.
 * Example 9:** Two similar right triangle prisms are below. If the ratio of the volumes is 343:125, find the missing sides in both triangles.
 * Solution:** The scale factor is 7:5, the cubed root. With the scale factor, we can now set up several proportions.
 * Example 10:** The ratio of the surface areas of two similar cylinders is 16:81. What is the ratio of the volumes?
 * Solution:** First, find the scale factor. If we take the square root of both numbers, the ratio is 4:9. Now, cube this to find the ratio of the volumes, [[image:http://www.ck12.org/ck12/ucs/?math=4%5E3%20%3A%209%5E3%20%3D%2064%3A729 caption="4^3 : 9^3 = 64:729"]].

Review Questions
Determine if each pair of right solids are similar. Below are two similar square pyramids with a volume ratio of 8:27. The base lengths are equal to the heights. Use this to answer questions 15-18. Use the hemispheres below to answer questions 19-20.
 * Questions 1-4 are similar to Examples 1 and 2.
 * Questions 5-14 are similar to Examples 3-8 and 10.
 * Questions 15-18 are similar to Example 9.
 * Questions 19 and 20 are similar to Example 1.
 * 1) [[image:http://www.ck12.org/ck12/images?id=327897 width="190"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=327898 width="230"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=328019 width="200"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=328032 width="230"]]
 * 5) Are all cubes similar? Why or why not?
 * 6) Two prisms have a scale factor of 1:4. What is the ratio of their surface areas?
 * 7) Two pyramids have a scale factor of 2:7. What is the ratio of their volumes?
 * 8) Two spheres have radii of 5 and 9. What is the ratio of their volumes?
 * 9) The surface area of two similar cones is in a ratio of 64:121. What is the scale factor?
 * 10) The volume of two hemispheres is in a ratio of 125:1728. What is the scale factor?
 * 11) A cone has a volume of [[image:http://www.ck12.org/ck12/ucs/?math=15%20%5Cpi caption="15 pi"]]and is similar to another larger cone. If the scale factor is 5:9, what is the volume of the larger cone?
 * 12) The ratio of the volumes of two similar pyramids is 8:27. What is the ratio of their total surface areas?
 * 13) The ratio of the volumes of two tetrahedrons is 1000:1. The smaller tetrahedron has a side of length 6 cm. What is the side length of the larger tetrahedron?
 * 14) The ratio of the surface areas of two cubes is 64:225. What is the ratio of the volumes?
 * 1) What is the scale factor?
 * 2) What is the ratio of the surface areas?
 * 3) Find [[image:http://www.ck12.org/ck12/ucs/?math=h%2C%20x caption="h, x"]]and [[image:http://www.ck12.org/ck12/ucs/?math=y caption="y"]].
 * 4) Find the volume of both pyramids.
 * 1) Are the two hemispheres similar? How do you know?
 * 2) Find the ratio of the surface areas and volumes.

Keywords, Theorems, & Formulas

 * Exploring Solids**
 * Polyhedron
 * Face
 * Edge
 * Vertex
 * Prism
 * Pyramid
 * Euler’s Theorem
 * Regular Polyhedron
 * Regular Tetrahedron
 * Cube
 * Regular Octahedron
 * Regular Dodecahedron
 * Regular Icosahedron
 * Cross-Section
 * Net
 * Surface Area of Prisms & Cylinders**
 * Lateral Face
 * Lateral Edge
 * Base Edge
 * Right Prism
 * Oblique Prism
 * Surface Area
 * Lateral Area
 * Surface Area of a Right Prism
 * Cylinder
 * Surface Area of a Right Cylinder
 * Surface Area of Pyramids & Cones**
 * Surface Area of a Regular Pyramid
 * Cone
 * Slant Height
 * Surface Area of a Right Cone
 * Volume of Prisms & Cylinders**
 * Volume
 * Volume of a Cube Postulate
 * Volume Congruence Postulate
 * Volume of a Rectangular Prism
 * Volume of a Prism
 * Cavalieri’s Principle
 * Volume of a Cylinder
 * Volume of Pyramids & Cones**
 * Volume of a Pyramid
 * Volume of a Cone
 * Surface Area and Volume of Spheres**
 * Sphere
 * Great Circle
 * Surface Area of a Sphere
 * Volume of a Sphere
 * Extension: Similar Solids**
 * Similar Solids
 * Surface Area Ratio
 * Volume Ratio

Review Questions
Match the shape with the correct name. Match the formula with its description.
 * 1) Triangular Prism
 * 2) Icosahedron
 * 3) Cylinder
 * 4) Cone
 * 5) Tetrahedron
 * 6) Pentagonal Prism
 * 7) Octahedron
 * 8) Hexagonal Pyramid
 * 9) Octagonal Prism
 * 10) Sphere
 * 11) Cube
 * 12) Dodecahedron
 * 1) Volume of a Prism - A. [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%7B3%7D%20%5Cpi%20r%5E2%20h caption="frac{1}{3} pi r^2 h"]]
 * 2) Volume of a Pyramid - B. [[image:http://www.ck12.org/ck12/ucs/?math=%5Cpi%20r%5E2%20h caption="pi r^2 h"]]
 * 3) Volume of a Cone - C. [[image:http://www.ck12.org/ck12/ucs/?math=4%20%5Cpi%20r%5E2 caption="4 pi r^2"]]
 * 4) Volume of a Cylinder - D. [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B4%7D%7B3%7D%20%5Cpi%20r%5E3 caption="frac{4}{3} pi r^3"]]
 * 5) Volume of a Sphere - E. [[image:http://www.ck12.org/ck12/ucs/?math=%5Cpi%20r%5E2%2B%20%5Cpi%20rl caption="pi r^2+ pi rl"]]
 * 6) Surface Area of a Prism - F. [[image:http://www.ck12.org/ck12/ucs/?math=2%20%5Cpi%20r%5E2%2B2%20%5Cpi%20rh caption="2 pi r^2+2 pi rh"]]
 * 7) Surface Area of a Pyramid - G. [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%7B3%7D%20Bh caption="frac{1}{3} Bh"]]
 * 8) Surface Area of a Cone - H. [[image:http://www.ck12.org/ck12/ucs/?math=Bh caption="Bh"]]
 * 9) Surface Area of a Cylinder - I. [[image:http://www.ck12.org/ck12/ucs/?math=B%2B%5Cfrac%7B1%7D%7B2%7D%20Pl caption="B+frac{1}{2} Pl"]]
 * 10) Surface Area of a Sphere - J. The sum of the area of the bases and the area of each rectangular lateral face.

Texas Instruments Resources

 * //In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See [].//**

Study Guide
Keywords: Define, write theorems, and/or draw a diagram for each word below. Polyhedron Face Edge Vertex Prism Pyramid Euler’s Theorem Regular Polyhedron Regular Tetrahedron Cube Regular Octahedron Regular Dodecahedron Regular Icosahedron Cross-Section Net Lateral Face Lateral Edge Base Edge Right Prism Oblique Prism Surface Area Lateral Area Surface Area of a Right Prism Cylinder Surface Area of a Right Cylinder Surface Area of a Regular Pyramid Cone Slant Height Surface Area of a Right Cone Volume Volume of a Cube Postulate Volume Congruence Postulate Volume Addition Postulate Volume of a Rectangular Prism Volume of a Prism Cavalieri’s Principle Volume of a Cylinder Volume of a Pyramid Volume of a Cone Sphere Great Circle Surface Area of a Sphere Volume of a Sphere Similar Solids Surface Area Ratio Volume Ratio = =
 * [[image:http://www.ck12.org/ck12/ucs/?math=1%5E%7Bst%7D caption="1^{st}"]]Section: Exploring Solids**
 * Homework:**
 * [[image:http://www.ck12.org/ck12/ucs/?math=2%5E%7Bnd%7D caption="2^{nd}"]]Section: Surface Area of Prisms & Cylinders**
 * Homework:**
 * [[image:http://www.ck12.org/ck12/ucs/?math=3%5E%7Brd%7D caption="3^{rd}"]]Section: Surface Area of Pyramids & Cones**
 * Homework:**
 * [[image:http://www.ck12.org/ck12/ucs/?math=4%5E%7Bth%7D caption="4^{th}"]]Section: Volume of Prisms & Cylinders**
 * Homework:**
 * [[image:http://www.ck12.org/ck12/ucs/?math=5%5E%7Bth%7D caption="5^{th}"]]Section: Volume of Pyramids & Cones**
 * Homework:**
 * [[image:http://www.ck12.org/ck12/ucs/?math=6%5E%7Bth%7D caption="6^{th}"]]Section: Surface Area and Volume of Spheres**
 * Homework:**
 * Extension: Similar Solids**
 * Homework:**

= = =Surface Area=

A prism is a polyhedron with two congruent, parallel bases. Click on picture to go to video explaining surface area of prism. A Pyramid is a polyhedron in which one face is can be any polygon and the other faces are triangles that meet at a common vertex. Click on picture to go to a video explaining surface area of a pyramid. A cone is like a pyramid but its base is a circle. Click on the picture to go to a video explaining surface area of a cone.