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## High school geometry

### Course: High school geometry>Unit 9

Lesson 2: Cavalieri's principle and dissection methods

# Volume of a pyramid or cone

Where does the 1/3 come from in the formula for the volume of a pyramid? How does the volume of a cone relate? What about oblique pyramids (pyramids that lean to the side)?

## What are pyramids and cones?

A pyramid is the collection of all points (inclusive) between a polygon-shaped base and an apex that is in a different plane from the base.
Another way to think of a pyramid is as a collection of all of the dilations of the base with the apex as the center of dilation with scale factors from $0$ to $1$.
A cone is a common pyramid-like figure where the base is a circle or other closed curve instead of a polygon. A cone has a curved lateral surface instead of several triangular faces, but in terms of volume, a cone and a pyramid are just alike.

## Volume of a pyramid

The formula for the volume $V$ of a pyramid is $V=\frac{1}{3}\left(\text{base area}\right)\left(\text{height}\right)$. Where does that formula come from?

### Where does the $\frac{1}{3}$‍  come from in the formula?

Suppose we start with a cube with a side length of $1$ unit. We can slice that cube into $3$ congruent pyramids.
Problem 1
What is the volume of each pyramid?
cubic units

### Scaling the pyramid

Scaling a pyramid works exactly the same way as scaling the prism that encloses it does. When we scale a pyramid with volume ${V}_{\text{unscaled}}$ by factors of $r$, $s$, and $t$ in three perpendicular directions, then the volume ${V}_{\text{scaled}}$ of the scaled figure is ${V}_{\text{unscaled}}rst$.
Problem 2
The following pyramid is a scaled version of the previous square-based pyramid, using different scale factors for each dimension.
What is the volume of the rectangle-based pyramid?
${\text{cm}}^{3}$

Key idea: The volume of pyramid is still $\frac{1}{3}$ of the volume of the prism that encloses it, even after we scale them both.

### Sliding the slices

Imagine we sliced the pyramid into layers parallel to its base. We could slide those layers without changing the volume. As the number of layers gets close to infinity, our reshaped pyramid smooths out.
Cavalieri's principle says that as long as we don't change the height or the areas of the pyramid's cross-sections parallel to its base, we don't change the volume either! We can use the same formula for pyramid volume no matter where we move the apex.
Problem 3
What is the volume of the pyramid?
${\text{m}}^{3}$

### Changing the base shape

There's another really fascinating application of Cavalieri's principle to pyramids. Two bases can have the same area and entirely different shapes. If the height and base area of two pyramids or
solids are equal, then so are their volumes, because the areas of all of the other cross-sections parallel to the base must be equal too.
So our formula ${V}_{\text{pyramid}}=\frac{1}{3}\left(\text{base area}\right)\left(\text{height}\right)$ works, no matter what 2D shape the base has.
Problem 4.1
The following pyramid has an isosceles right triangle as a base.
What is the volume of the pyramid?
${\text{cm}}^{3}$

### Getting $\frac{1}{3}$‍  another way

Another way that mathematicians like you have convinced themselves that the volume of a pyramid is $\frac{1}{3}$ the volume of the prism that encloses it is by approximating the volume using prisms.
We can model a pyramid as a stack of prisms, like building a pyramid out of blocks. This model has a volume that is a greater than the pyramid's volume. As we get more, thinner layers, we get closer and closer to the volume of the pyramid.
Number of layers$\frac{\text{Volume of block pyramid approximation}}{\text{Volume of prism}}$
$4$$\approx 0.469$
$16$$\approx 0.365$
$64$$\approx 0.341$
$256$$\approx 0.335$
$1024$$\approx 0.334$
$4096$$\approx 0.333$
$\mathrm{\infty }$$\frac{1}{3}$
Since prism-like figures can have any closed, 2D figure for the base, and since we can slide the prisms without changing their volume, the ratio holds true for all pyramid-like figures, including cones.