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.

### Course: Geometry (FL B.E.S.T.)>Unit 9

Lesson 5: 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 $\text{height}$ and the same $\text{cross-sectional area}$ at every point along that height, they have the same $\text{volume}$.

## 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 $\text{height}$ of $21$ and a $\text{base area}$ of $64\pi$.
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
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?
• The volume for a cone and pyramid are the same, V = 1/3 Bh where B is the area of the base. So even though the base is a different shape, as long as the areas and heights are the same, they will have the same volume. Just because the way to find the areas differ by the shape of the base, it does not change the area of the bases. Of course, it would be hard to get a circle and a pentagon to have the exact same area because this would mean the side of pentagon would have to be a multiple of π.
• The last statement on this article i believe is incorrect. Suppose you have a cylinder with a base area of 5 and a height of three and a cone of a base area of 5 and a height of 3 they will have a different volume. The principle should work with the same type of shapes and not completely different shapes as they all have different volume formulas.
• Cavalieri's wouldn't apply to a cone and cylinder because while the base of each might be equal, you could not take cross-sections at an arbitrary height and show them to be equal. This is why the language of Cavalieri's Principle is specific to areas of cross-sections and not just areas of bases.

Now as long as the areas of cross-sections are equal then the shape of those cross-sections is irrelevant. This is actually one of the ways in which you can derive the formula for the volume of a sphere... it is equivalent to a cylinder that has been hollowed out by cones (think inverse of an hourglass). It compares areas of circles to areas of rings.
• I’m on mobile, so not many options, how do I express pi.
• Sorry, you'll have to do a decimal. :|
• In the problem 2.1, I still cannot understand why these shapes can have the same volume. Is the volume of the pentagon-based pyramid the same as the other 2 cones even though they use pi?
(1 vote)
• Yes, the same volume.

Like you said, to calculate the cones, we have to use pi, because:
to calculate the base, which is a circle, area=pi*r^2

But the problem said: “The following figures all have the same height and same base area.”.
Let’s make the area of the circle equals pi, so the area of pentagon is also pi.
As we know, volume = 1/3 * area * height, (or the Cavalieri’s principle)
plus, they share the same area & height,
so the volumes are the same.

(English is not my first language and I hope it helps you.)