Session 8, Part B:
Volume Formulas

In This Part: Cross-Section Method | Cylinders, Cones, and Spheres

Many of the three-dimensional solids you encounter in everyday life are not in the shape of a prism or a cylinder. For example, cones and spheres are common shapes. There are a number of ways to approximate the volume of these solids. Note 4
In this next activity, you will compare the volumes of a cylinder, cone, and sphere which all have the same radius and the same height.

 • The relevant dimensions of a cone are its height and the radius of its circular base. • A sphere is described by its radius (the height of a sphere is simply its diameter). • A cylinder is described by the radius of a circular base and its height.

This activity requires the Flash plug-in, which you can download for free from Macromedia's Web site. This Interactive Activity simulates a method for exploring the relationships among the volumes of these shapes. You can read about these methods in the non-interactive version of this activity, and try them hands-on if you wish.

 Problem B4 What is the relationship between the volume of the sphere and the volume of the cylinder?

 Try to avoid as much measurement error as possible by lining up the top of the sphere and the top of the cylinder.    Close Tip Try to avoid as much measurement error as possible by lining up the top of the sphere and the top of the cylinder.

 Problem B5 What is the relationship between the volume of the cone and the volume of the cylinder?

 Sometimes this method of determining the relationship between the volume of a cone and a cylinder is not very accurate because the cone does not hold its shape.   Close Tip Sometimes this method of determining the relationship between the volume of a cone and a cylinder is not very accurate because the cone does not hold its shape.

Alternate Experiment

 • Take a plastic cone, sphere, and cylinder with the same height and radius. Using water or rice, experiment with filling the solids to determine relationships among their volumes. If your plastic solids are small, fill with water for a more precise approximation. Larger models are easier to work with and can be filled with either material.

 Problem B6 If a cone, cylinder, and sphere have the same radius and the same height, what is the relationship among the volumes of the three shapes?

Problem B7

Using the illustration above, write the formulas you could use to find the volume of the following:

 a. a cylinder

 Use the cross-section method. Remember, the height is twice the radius in this case.   Close Tip Use the cross-section method. Remember, the height is twice the radius in this case.

 b. a cone

 Use the formula for a cylinder and what you know about the ratios.   Close Tip Use the formula for a cylinder and what you know about the ratios.

 c. a sphere

 Use the formula for a cylinder and what you know about the ratios.   Close Tip Use the formula for a cylinder and what you know about the ratios.

 How are the formulas connected to your physical discoveries?

Problem B8

Are there similar relationships between other three-dimensional solids such as rectangular prisms and pyramids? In this final activity, compare the volumes of
pairs of solids (PDF file). Record what is the same for both solids (e.g., height) and note how the volumes of the two solids are related. Try to generalize the relationships among volumes for similar three-dimensional solids. Fill in the table. Note 6

Pair

Solids

What's the same?

How are the volumes related?

 1 A, B 2 C, D 3 E, F 4 G, H 5 I, J 6 I, K

Pair

Solids

What's the same?

How are the volumes related?

1

A, B

 Same base and height

 Volume B is one-third of A.

2

C, D

 Same base and height

 Volume D is one-third of C.

3

E, F

 Same base and height

 Volume F is one-third of E.

4

G, H

 Same base, height of H is twice the height of G

 Volume H is two-thirds of G.

5

I, J

 Same height, base J is half the area of base I

 Volume J is one-sixth of I.

6

I, K

 Same base and height

 Volume K is one-third of I.

 Problem B9 Based on your findings in the previous problem, can you make any generalizations about how the volumes of some three-dimensional solids are related?

 Problem B10 Write formulas for the volume of a square pyramid and a triangular pyramid. How are the volumes of pyramids and cones related?

 Video Segment Boston's Big Dig is the most expensive public works project in the history of the United States. In this segment, Michael Bertoulin explains how engineers calculate the volume of irregular shapes by breaking them down into smaller, regular shapes. As you'll see, calculating volume is only one in a series of engineering and technological challenges engineers have to overcome. If you are using a VCR, you can find this segment on the session video approximately 22 minutes and 18 seconds after the Annenberg Media logo.