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High school geometry

Course: High school geometry > Unit 9

Lesson 2: Cavalieri's principle and dissection methods
Math
High school geometry
Solid geometry
Cavalieri's principle and dissection methods
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Volume of a pyramid or cone

Google Classroom
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.
A slanted pyramid where the base is a rectangle. The base is labeled base. The tip of the pyramid is labeled apex.
A slanted pyramid where the base is a triangle. The base is labeled base. The tip of the pyramid is labeled apex.
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 pentagonal pyramid and a slanted pentagonal pyramid. Both have an equal number of cross sections.
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.
An upside down cone.

Volume of a pyramid

The formula for the volume V of a pyramid is V=13(base area)(height). Where does that formula come from?

Where does the 13 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?
  • Your answer should be
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
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 Vunscaled by factors of r, s, and t in three perpendicular directions, then the volume Vscaled of the scaled figure is Vunscaledrst.
A rectangular prism. Three arrows extend from the rectangular prism, one from the length, one from the height, and one from the width of the base.
A rectangular pyramid. Three arrows extend from the pyramid, one from apex and one each from the length and width of the base.
Problem 2
The following pyramid is a scaled version of the previous square-based pyramid, using different scale factors for each dimension.
An oblique rectangular pyramid with a face of the pyramid being a right triangle. This face has a base of six centimeters and a height of seven point five centimeters. Its rectangular base has a length of six centimeters and a width of two centimeters.
What is the volume of the rectangle-based pyramid?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
cm3

Key idea: The volume of pyramid is still 13 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.
A series of six rectangular pyramids. The first pyramid is an oblique rectangular pyramid where the top of the pyramid leans left. It has been cut into four sections, which have been rearranged in a less leaning fashion in the second pyramid. From left the right, the pyramids have more sections that are parallel to the base. The final pyramid has infinitely many cross section so that it appears like smooth right rectangular pyramid.
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?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
m3

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 Vpyramid=13(base area)(height) works, no matter what 2D shape the base has.
Problem 4.1
The following pyramid has an isosceles right triangle as a base.
A pyramid with a base that is a right triangle with its legs both being fourteen centimeters. It has a vertical height of twenty-seven centimeters. The edge of the triangular faces that connect to the hypotenuse and the side of its right triangular base has a length of twenty-nine centimeters.
What is the volume of the pyramid?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
cm3

Getting 13 another way

Another way that mathematicians like you have convinced themselves that the volume of a pyramid is 13 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 layersVolume of block pyramid approximationVolume of prism
4≈0.469
16≈0.365
64≈0.341
256≈0.335
1024≈0.334
4096≈0.333
∞13
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.

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