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## Get ready for AP® Calculus

### Unit 7: Lesson 8

Cavalieri's principle and dissection methods# Cavalieri's principle in 2D

CCSS.Math:

If two figures have the same height and the same width at every point along that height, they have the same area.

## Cavalieri's principle in 2D

**Key idea:**If two 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 #a75a05, start text, w, i, d, t, h, end text, end color #a75a05 at every point along that height, they have the same start color #208170, start text, a, r, e, a, end text, end color #208170.

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According to

**Cavalieri's principle**, these figures have the same area because they have the same height left parenthesis, 6, right parenthesis and the same width left parenthesis, 4, right parenthesis as each other at every point along that height.## Cavalieri's principle with varied widths

Sometimes a figure has different widths at different heights. Cavalieri's principle still works.

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## Want to join the conversation?

- How can we be sure that two figures have the same width at every point along the height?(14 votes)
- Typically, it will tell you the width if you need it. For example, in a cylinder, the diagram/problem will usually state the radius or diameter. That is the same as the width. This also works for other shapes.(4 votes)

- maybe it would help more by putting the formila to the problem(10 votes)
- how does pi affect a wavy rectangle(1 vote)
- it doesn't the length can be an arbitrary constant here pi is being treated as a constant and not significant definations in geometry(7 votes)

- I find the diagram of the last question confusing. I took 3pi to be the distance from the beginning of the wavy figure to the end, rather than the length of the wavy edge.(2 votes)
- That's porbably because 3pi
*is*the distance from the beginning of the wavy figure to the end, rather than the length of the wavy edge. The notation, with a number line with vertical lines, is generally assumed to mean the length of the figure. If 3pi was the length of the edge, such a detail would be stated.(2 votes)

- I thought the last one was a 3D figure and I got confused(1 vote)