If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## High school geometry

### Unit 9: Lesson 2

Cavalieri's principle and dissection methods

# Cavalieri's principle in 2D

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.
Wait a second, you already know this! Check it out.
Rectangle with a base of 4 units and height of 6 units.
What is the area of the rectangle?

Parallelogram with a base of 4 units and height of 6 units.
What is the area of the parallelogram?

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.
According to Cavalieri's principle, which of the following figures must have the same area?

## Try this

What is the area of the following figure?

## Want to join the conversation?

• How can we be sure that two figures have the same width at every point along the height?
• 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.
• maybe it would help more by putting the formila to the problem