Geometry

Chapter 11Section 2 & 3: Volume of Prisms, Cylinders, Pyramids, and Cones

The ______of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior.

Cavalieri's Principle: If two three-dimensional figures have the same height and have the same cross-sectional area at every level, they have the same volume.

Example 1: Finding Volumes of Prisms

Find the volume of each prism. Round to the nearest tenth, if necessary.

a. b. apothem = 3ft

c. Find the volume of a triangular prism with a height of 9 yd whose base is a right triangle with legs 7 yd and 5 yd long.

d. A swimming pool is a rectangular prism. Estimate the volume of water in the pool in gallons when it is completely full (Hint: 1 gallon ≈ 0.134 ft3). The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds.

Example 2: Finding Volumes of Cylinders

Cavalieri’s principle also relates to cylinders. The two stacks have the same number

of CDs, so they have the same volume.

Find the volume of each cylinder. Give your answers both in terms ofand rounded to the nearest tenth.

a.

Example 3: Exploring Effects of Changing Dimensions

a. The length, width, and height of the prism are doubled. Describe the effect on the volume.

b. The radius and height of the cylinder are multiplied by. Describe the effect on the volume.

Example 4: Finding Volumes of Composite Three-Dimensional Figures

a. Find the volume of the composite figure. Round to the nearest tenth.

b. Find the volume of the composite figure. Round to the nearest tenth.

Example 5: Finding volumes of pyramids

a. Find the volume a rectangular pyramid with

length 11 m, width 18 m, and height 23 m.

Example 6: Finding volumes of cones

Find the volume of each cone. Give your answers both in terms ofand rounded to the nearest tenth.

a. Find the volume of a cone with radius 7 cm and height 15 cm. Give your answers both in terms of  and rounded to the nearest tenth.

b. c.

d. Find the volume of the composite figure. Round to the nearest tenth.

Example 7: Exploring the Effects of Changing Dimensions

Describe the effect on the volume.

a. The radius and height of the cone are doubled. b. The edge length of a cube is tripled.

Example 8: Comparing Surface Area and Volume.

Suggest three different sets of dimensions that you could use to build a rectangular prism that has a volume of 8 cubic units. Choose one set and draw the net of your rectangular prism. Then create a 3-dimensional sketch of your rectangular prism. Label the length, the width, and the height. Finally, find the surface area of your rectangular prism.

Surface area = ______

Do you think all rectangular prisms with a volume of 8 cubic units have the same surface area? Explain.