In the previous section, you looked for relationships between the volume formulas for prisms and pyramids. In this section, you will investigate relationships between cylinders, cones, and spheres.

Volume of Cylinders

This activity might not be viewable on your mobile device. Interactive exercise. Assistance may be required. Click on the image below to access the interactive to investigate the formula for the volume of cylinders. Follow the onscreen prompts until you reach the summary screen in the interactive.

NeedInteractive popup. Assistance may be required.additional directions ?

What was the area of the circular base of the cylinder?

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Check Your Answer

A=πr² = 3.14 x 2² = 3.14 x 4 = 12.56 square inches

What was the height of the height of the cylinder?

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Check Your Answer

6 inches

How did you determine the area of the base?

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Check Your Answer

One way to determine the area is to use the area formula for a circle A = πr²

What was the volume of the cylinder?

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Check Your Answer

: V=πr² h = 12.56 sq in×6 in = 75.36 cubic inches

How is the volume of a cylinder related to the area of the base of the cylinder and the height of the cylinder?

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Check Your Answer

The volume of a cylinder is equal to the product of the area of the base of the cylinder and the height of the cylinder.

Interactive exercise. Assistance may be required. In the following interactive, drag the appropriate symbols into the appropriate place to build a formula for calculating the volume of any cylinder. You may use the tiles more than once.

Volumes of Cones and Spheres

Cones and spheres have a special relationship to cylinders. As with pyramids and prisms, let’s begin by thinking about a cylinder, cone, and sphere with congruent dimensions:

Interactive exercise. Assistance may be required. The animation below shows the same cylinder and cone from the diagram. Use the animation to see how many times you need to fill the cone and pour it into the cylinder before you have a completely full cylinder.

How many times did you need to fill the cone and pour it into the cylinder in order to completely fill the cylinder?

Interactive popup. Assistance may be required.Check Your Answer3 times

If the formula for the volume of a cylinder is V = Bh, what is the formula for the volume of a cone?

Interactive popup. Assistance may be required.Check Your AnswerV = 1 over 3 1 3Bh

If the formula for the volume of a cylinder is V = πr²h, what is the formula for the volume of a cone?

Interactive popup. Assistance may be required.Check Your AnswerV = 1 over 3 1 3πr²h

Interactive exercise. Assistance may be required. The animation below shows the same cone and sphere from the diagram. Use the animation to see how many times you need to fill the cone and pour it into the sphere before you have a completely full sphere.

How many times did you need to fill the cone and pour it into the sphere in order to completely fill the sphere?

Interactive popup. Assistance may be required.Check Your Answer2 times

If the formula for the volume of a cone V = 1 over 3 1 3πr²h, what is the formula for the volume of a sphere?

Interactive popup. Assistance may be required.Check Your AnswerV = 2 x1 over 3 1 3πr²h = 2 over 3 2 3πr²h

If the “height” of the sphere is equal to the diameter (2 times the radius, or 2r), how does that change your volume formula for a sphere?

Interactive popup. Assistance may be required.Check Your AnswerV = 2 over 3 2 3πr² (2r) = 4 over 3 4 3πr³

Practice

Interactive exercise. Assistance may be required. Match each figure below with an expression that could be used to determine its volume. Move the expression to see the original formula from which it comes.