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## Get ready for AP® Calculus

### Course: Get ready for AP® Calculus>Unit 7

Lesson 8: Cavalieri's principle and dissection methods

# Cavalieri's principle in 3D

If two 3D figures have the same height and the same cross-sectional area at every point along that height, they have the same volume.

## Cavalieri's principle in 3D

Key idea: If two 3D figures have the same start color #7854ab, start text, h, e, i, g, h, t, end text, end color #7854ab and the same start color #208170, start text, c, r, o, s, s, negative, s, e, c, t, i, o, n, a, l, space, a, r, e, a, end text, end color #208170 at every point along that height, they have the same start color #ca337c, start text, v, o, l, u, m, e, end text, end color #ca337c.

## Why it works

Imagine we have a stack of coins (or books, playing cards, or anything with parallel planes). If we push on the top of the stack so that it slopes to the side, have we changed the volume? Of course not!
We can cut a solid into many parallel layers, then slide them from side to side without changing the volume.
Try out the Cavalieri's principle cylinder simulation for yourself. Drag the sliders to change the number slices and how far the cylinder on the left is skewed. Try increasing the number of slices until the cylinder looks smooth.

## Exploring more unusual shapes

We can use Cavalieri's principle for more than just prisms and cylinders. For example, we can slide the layers of a cone from side to side without changing the volume, too.
Try the Cavalieri's sculpture simulation for yourself. Drag your mouse over the cone on the right to sculpt it. Notice that no matter how you sculpt, the cross-sectional areas of both figures at any given height remain equal.
Both figures have a start color #7854ab, start text, h, e, i, g, h, t, end text, end color #7854ab of start color #7854ab, 21, end color #7854ab and a start color #208170, start text, b, a, s, e, space, a, r, e, a, end text, end color #208170 of start color #208170, 64, pi, end color #208170.
Problem 1
What is the volume of cone on the left?
cubic units
What is the volume of sculpted cone on the right?
cubic units

## Cavalieri's principle with different shapes

One of the more useful features of Cavalieri's principle is that it works even when the cross-sections have different shapes, as long as they still have equal areas.
Problem 2.1
• Current
The following figures all have the same height and same base area.
Which of the following figures have the same volume?

## Want to join the conversation?

• How does the last option on the 1st question, the one with the hexagon as a base, have the same volume as the other two?