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?

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

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?

What is the volume of a cone with a base radius of 3 units and a height of 4 units? Use the formula for the volume of a cone $$V = \frac{1}{3}\pi r^2h$$ .

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?

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?

If a solid has a constant cross-sectional area of $$A$$ square units along its height $$h$$ , what is the formula for its volume?

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?

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?

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