Cross Sections and Volume

Cross Sections and Volume

Let’s look at how cross sections and volume are related.

I know that if two solids have equal-area cross sections at all heights, they have the same volumes.

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​Learning Target

​Cavalieri’s Principle

If two solids with the same 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.

​oblique (solid)

Prisms and cylinders are said to be oblique if when one base is translated to coincide with the other, the directed line segment that defines the translation is not perpendicular to the bases.

A cone is said to be oblique if a line drawn from its apex at a right angle to the plane of its base does not intersect the center of the base. The same definition applies to pyramids whose bases are figures with a center point, such as a square or a regular pentagon.

​right (solid)

Prisms or cylinders are said to be right if when one base is translated to coincide with the other, the directed line segment that defines the translation is perpendicular to the bases.

A cone is said to be right if a line drawn from its apex at a right angle to the plane of its base passes through the center of the base. The same definition applies to pyramids whose bases are figures with a center point, such as a square or a regular pentagon.

Open Ended

The images show the same number of coins arranged in different ways.

How are the two coin stacks different from each other?

Open Ended

The images show the same number of coins arranged in different ways.

What geometric solid does either stack of coins resemble ? Which Stack?

Open Ended

The images show the same number of coins arranged in different ways.

How do the heights of the two stacks compare?

Open Ended

The images show the same number of coins arranged in different ways.

How do the volumes of the two stacks compare? Explain your reasoning.

https://youtu.be/2TDFtCKMoKk​

https://www.geogebra.org/m/ptscstvq

Poll

For each pair of solids, select the pair that you have determined the volumes of the two solids are equal.

Open Ended

Calculate the volume of this cylinder in terms of Pi and to the nearest hundredth