Cavalieri's Principles

A right circular cylinder and an oblique circular cylinder are given.
True or False? If the radii of both cylinders are equal, the volumes of the cylinders will be equal because they have the same height.

• True

• False

2. A right circular cylinder and a right rectangular prism are given.
   True or False? Cavalieri's Principle does not apply to these solids because their bases are not the same shape.

• True
• False

3. The three solids shown above have the same height and matching cross sectional areas parallel to the bases. Which of the solids have the same volume?

• A and C only

• B and C only

• A and B only

• A, B and C

A right circular cylinder, A, and an oblique circular cylinder, B, are shown at the right. Find the volume of cylinder B in cubic inches.

• 110π in³

• 220π in³

• 440π in³

• 550π in³

By Cavalieri's Principle, this right triangular prism and right rectangular prism have the same volume. If the center plane intersects the solids parallel to their bases, which of the following choices could be the base and height of the triangular cross section?

• h = 4; b = 12

• h = 4; b = 8

• h = 8; b = 12

• h = 8; b = 14

What is the volume of an oblique cylinder with a radius of 3 and a height of 9?

• 27 units³

• 81 units³

• 81π units³

• 27π units³

The right circular cylinder and the right rectangular prism have the same heights and the same base areas. A plane, parallel to the bases slices the two solids. If the rectangular cross section has a base length of 16 inches and a width of 4π inches, what is the radius of the cylinder's cross section?

• 32 in.

• 8π in.

• 8 in.

• 6π in.

A series of coins are stacked to represent a right circular cylinder (on the left). The coins are then "slid" to represent a distorted cylinder (on the right). The same number of congruent coins was used in each stack.

Which of the following statements will be TRUE regarding these stacks of coins?

• The volume of both stacks will be the same.

• The area of a cross section parallel to the bases will not be equal due to the distorted nature of the second stack.

• The height of the distorted stack will be slightly larger than that of the straight stack.

• Cavalieri's Principle can be used in this situation to verify that the volumes of the stacks are equal.

Two solids have equal heights. If every cross-section taken parallel to their bases is congruent, which principle guarantees they have the same volume?

• Archimedes Principles

• Pythagorean Theorem

• Cavalieri's Principles

• Euler's Formula

What is the necessary condition for two solids to be compared using Cavalieri's Principle?

• Identical Shapes

• Equal Volumes

• Equal height an congruent cross section

• Equal base area