Lesson 5: Cavalieri's Principle

Two solids have the same height. At every level, the cross-sectional area of the first solid is equal to the cross-sectional area of the second solid. According to Cavalieri's Principle, which of the following statements is true?

• The two solids have different volumes.

• The two solids have the same volume.

• The first solid has a larger volume than the second solid.

• The second solid has a larger volume than the first solid.

A cylinder and a prism stand on equal bases and have the same height. Which of the following statements is correct regarding their volumes?

• The cylinder has a larger volume because it is round.

• The prism has a larger volume because it has flat sides.

• They have the same volume.

• It is impossible to determine without knowing the specific shapes.

If two solids are composed of layers of material, each of identical thickness, and the cross-sectional area of one solid at any height is equal to the cross-sectional area of the other solid at the same height, what can be said about their volumes?

• The volumes can be different.

• The volumes are the same.

• The first solid has twice the volume of the second.

• The second solid has half the volume of the first.

Two pyramids have the same height. The base area of the first pyramid is 20 square units, and the base area of the second pyramid is 30 square units. What is the ratio of their volumes?

• 2:3

• 1:1

• 3:2

• 1:2

A swimming pool is being filled with water. The cross-sectional area of the pool at every depth is the same. If the pool is filled to a depth of 2 meters, and the cross-sectional area is 10 square meters, what is the volume of water in the pool?

• 5 cubic meters

• 10 cubic meters

• 20 cubic meters

• 40 cubic meters

How does Cavalieri's Principle apply to comparing the volume of a sphere with a radius of 3 units to a cylinder with the same height and base radius?

• The sphere has a larger volume because it is a perfect shape.

• The cylinder has a larger volume because it has a larger base area.

• The sphere and the cylinder have the same volume.

• The sphere has two-thirds the volume of the cylinder.

A solid has a height of 10 units. At a height of 5 units from the base, the cross-sectional area is 4 square units. If the cross-sectional area remains constant throughout the height, what is the volume of the solid?

• 20 cubic units

• 40 cubic units

• 60 cubic units

• 80 cubic units

Using Cavalieri's Principle, how can you determine if two cylinders have the same volume?

• Compare their heights.

• Compare their base radii.

• Ensure they have the same height and base area.

• Measure the water displacement when submerged.