Volume of a Frustum of Pyramids & Cones

Volume of a Frustum of Pyramids & Cones
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  • 0:03 What Are Frustums?
  • 0:43 Similar Triangles
  • 2:27 Frustum Volume in a Cone
  • 5:47 The Giza's Frustum's Volume
  • 6:52 Extending the Equation…
  • 8:00 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Knowing how to calculate the volume of a cone or a pyramid leads to equations for the volume of a frustum. In this lesson, we derive two equations for calculating the volume of frustums.

What Are Frustums?

The Great Pyramid at Giza is missing its capstone top. How would we calculate the volume of this pyramid?

When the top part of a pyramid or cone has been cut off along a plane parallel to its base, the remaining portion is called a frustum. The top portion that's been removed is itself a pyramid or a cone.

The volume of a frustum is easy to find when we know the heights of the original pyramid and the smaller pyramid that's been removed, but what if we don't know those heights? We'll have the answer to that in a moment, but to understand the equations for calculating volumes of frustums of pyramids and cones, we must first start with a discussion of similar triangles.

Similar Triangles

When triangles are scaled versions of each other, they are similar triangles.

similar triangles

When two triangles are similar, the ratio of corresponding sides is the same. The ratios r2 : r1 and y - h : y are equal,

ratio_of_sides_in_similar_triangles

Consider r2 as the radius of a circle whose area A2 is π r22. Likewise, a circle whose radius is r1 has area A1 = π r12.

The ratio of the areas:

 ratio_of_two_areas

We're going to solve for y because it's going to show us how the height of the frustum relates to the height of the original cone or pyramid in our later equations for volume:

 square_root_of_both_sides

Now we switch the sides and multiply by y:

 multiplying_both_sides_by_y

Then transfer the h and y terms:

isolate_the_y

Then we factor, which gives:

factor_the_y

Then we divide:

divide_both_sides_by_the_y_factor

Then we multiply the numerator and the denominator by the square root of A1:

similar triangle replacement

Cool, now we can see that y is isolated.

Frustum Volume in a Cone

What about cones?

Let's take three shapes:

  • A cone of height y
  • A frustum of height h
  • A smaller cone of height y - h

Take the cone of height y and slice off a piece. The result is a frustum of height h. The sliced piece is also a cone, but its height is y - h. Make sense? Placing the sliced piece back onto the frustum gives the original cone, so the total height would be y - h + h = y, the height of the original cone.

The frustum volume is the volume of the original cone minus the volume of the sliced portion. Here's the frustum volume equation we will derive:

 volume_of_a_frustum

Notice that this equation allows us to find the volume of a frustum without knowing the original height y of the cone or pyramid. So, let's get to the bottom of this equation.

Generally, volumes of pyramids and cones are easy to calculate. Multiply 1/3 times the area of the base times the height. The height is a line perpendicular to the base reaching the peak.

The large cone has an area A1 and a height y. Thus, the volume V1 of the large cone:

The small cone has an area A2 and a height y - h. The volume V2 of the small cone is thus:

The frustum volume V is the large cone volume minus the small cone volume. Write V = V1 - V2 and substitute for V1 and V2:

Expanding the y - h term and collecting terms with a y:

A1 is the same as the square root of A1 times the square root of A1. Thinking like this, A1 - A2 is the difference of squares. Just like c2 - d2 is a difference of squares, which can be factored as (c - d)(c + d), factor the difference of A1 and A2:

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