Sign in to view assessments and invite other educators
Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.
The images show the same number of coins arranged in different ways.
A
B
The image shows two rectangular prisms. The bases of the prisms are congruent. Each base has an area of square units, and the prisms are the same height. A plane intersects the two prisms parallel to their bases, creating cross-sections.
For each pair of solids, decide whether the volumes of the two solids are equal. Explain your reasoning. If you and your partner disagree, discuss each other’s approach until you reach agreement.
Suppose we have a stack of paper in the shape of a rectangular prism. Then we shift the paper so the prism slants to the side. The first is called a right prism because its sides are at right angles to its base. The second one, without right angles between the sides and the base, is called an oblique prism. The volume of the prism doesn’t change when we shift it—the amount of paper stays the same.
In fact, Cavalieri’s Principle says that if any two solids are cut into cross-sections by parallel planes, and the corresponding cross-sections at all heights all have equal areas, then the solids have the same volume.
The oblique cylinder and the right prism in this image have equal volumes because they have the same height and their cross-sections at all heights have the same area of 6 square units.
These two pyramids also have equal volumes. Their bases are congruent, and they have the same height. Shifting a solid from right to oblique doesn’t change its volume.
A pyramid with a square base. The sides of the base are labeled 5. The height, inside the pyramid, is labeled 7. Another pyramid, slanted to the right, with square base. The sides of the base are labeled 5. The height, outside the pyramid, is labeled 7.
Cavalieri’s Principle states that if two solids of equal height are cut into cross-sections by parallel planes, and the corresponding cross-sections on each plane always have equal areas, then the two solids have the same volume.
An oblique solid is not exactly upright—it seems to lean over at an angle.
A right solid is exactly upright—it does not seem to lean over at an angle.