How to find the volume of a prism, cylinder, pyramid, cone, and sphere. Chapter (Volume)GeometryStandard/Goal 2.2

1. Check and discuss assignment from Friday. 2. Read, write, and discuss how to find the volume of a prism. 3. Read, write, and discuss how to find the volume of a cylinder. 4. Read, write, and discuss how to find the volume of a pyramid. 5. Read, write, and discuss how to find the volume of a cone. 6. Read, write, and discuss how to find the volume of a sphere. 7. Work on assignment.

Volume of a solid is the number of cubic units contained in its interior. The space that a figure occupies.

If two space figures have the same height and the same cross-sectional area at every level, Then they have the same volume.

The volume V of a prism is V = Bh, Where B is the area of a base, h is the height

The volume V of a cylinder is V = Bh, or V = r²h Where B is the area of a base, h is the height, r is the radius of a base

Find the volume of the prism below. The area of the base B = w = 3  5 = 15. V = Bh Use the formula for volume. = 15 5Substitute 15 for B and 5 for h. = 75Simplify. The volume of the rectangular prism is 75 in. 3. Lesson 11-4

Find the volume of the prism below. The prism is a right triangular prism with triangular bases. The base of the triangular prism is a right triangle where one leg is the base and the other leg is the altitude – 20 2 = 841  400 = 441  21 Use the Pythagorean Theorem to calculate the length of the other leg. Lesson 11-4

The volume of the triangular prism is 8400 m 3. The area B of the base is bh = (20)(21) = 210. Use the area of the base to find the volume of the prism V = Bh Use the formula for the volume of a prism. = Substitute. = 8400Simplify. Lesson 11-4 (continued)

Find the volume of the cylinder below. Leave your answer in terms of. The volume of the cylinder is 576 ft 3. V = r 2 h Use the formula for the volume of a cylinder. The formula for the volume of a cylinder is V = r 2 h. The diagram shows h and d, but you must find r. r = d = = 8 2 9Substitute. = 576Simplify. Lesson 11-4

Find the volume of the composite space figure. You can use three rectangular prisms to find the volume. Each prism’s volume can be found using the formula V = Bh. Lesson 11-4

Volume of prism I = Bh = (14 4) 25 = 1400 Volume of prism II = Bh = (6 4) 25 = 600 Volume of prism III = Bh = (6 4) 25 = 600 Sum of the volumes = = 2600 The volume of the composite space figure is 2600 cm 3. Lesson 11-4 (continued)

The volume V of a pyramid is where B is the area of a base, h is the height. h B

The volume V of a cone is Where B is the area of the base, h is the height, r is the radius of the base. h r B

Find the volume of a square pyramid with base edges 15 cm and height 22 cm. Because the base is a square, B = = 225. V = Bh Use the formula for volume of a pyramid = (225)(22)Substitute 225 for B and 22 for h = 1650Simplify. The volume of the square pyramid is 1650 cm 3. Lesson 11-5

Find the volume of a square pyramid with base edges 16 m and slant height 17 m. The altitude of a right square pyramid intersects the base at the center of the square. Lesson 11-5

Because each side of the square base is 16 m, the leg of the right triangle along the base is 8 m, as shown below. Step 1: Find the height of the pyramid = 8 2  h 2 Use the Pythagorean Theorem. 289 = 64  h 2 Simplify. 225 = h 2 Subtract 64 from each side. h = 15Find the square root of each side. Lesson 11-5 (continued)

Step 2: Find the volume of the pyramid. = 1280Simplify. The volume of the square pyramid is 1280 m 3. V = Bh Use the formula for the volume of a pyramid = (16  16)15Substitute Lesson 11-5 (continued)

Find the volume of the cone below in terms of. r = d = 3 in V = r 2 h Use the formula for volume of a cone = (3 2 )(11)Substitute 3 for r and 11 for h = 33Simplify. Lesson 11-5 The volume of the cone is 33 in. 3.

An ice cream cone is 7 cm tall and 4 cm in diameter. About how much ice cream can fit entirely inside the cone? Find the volume to the nearest whole number. r = = 2 d2d2 V = r 2 h Use the formula for the volume of a cone V = (2 2 )(7)Substitute 2 for r and 7 for h Use a calculator. About 29 cm 3 of ice cream can fit entirely inside the cone. Lesson 11-5

The volume V of a sphere with radius r is: r

Find the volume of the sphere. Leave your answer in terms of. V = r 3 Use the formula for the volume of a sphere = 15 3 Substitute r = = = 4500Simplify. The volume of the sphere is 4500 cm 3. Lesson 11-6

The volume of a sphere is 1 in. 3. Find its surface area to the nearest tenth. Step 1: Use the volume to find the radius r. V = r 3 Use the formula for the volume of a sphere = r 3 Substitute = r 3 Solve for r = r Find the cube root of each side r Use a calculator. Lesson 11-6

Step 2: Use the radius to find the surface area. S.A. = 4 r 2 Use the formula for the surface area of a sphere. 4 ( ) 2 Substitute Use a calculator. To the nearest tenth, the surface area of the sphere is 4.8 in. 2. Lesson 11-6 (continued)

Kennedy, D., Charles, R., Hall, B., Bass, L., Johnson, A. (2009) Geometry Prentice Hall Mathematics. Power Point made by: Robert Orloski Jerome High School.