Deriving the Volume Formula for the Frustum of a Cone

Have you ever wondered about the formula to calculate the volume of a frustum of a cone? A frustum is a solid formed by slicing off the top of a cone with a plane parallel to the base. This article will guide you through the derivation of the volume formula for the frustum of a cone. Let's dive in!

Understanding the Frustum of a Cone

A frustum of a cone is the portion of a cone that remains after the top part has been cut off by a plane parallel to the base. If you extend the slant heights of the frustum's sides, it tapers to a point, forming two cones (the larger and the smaller).

Volume of the Frustum of a Cone

The volume of the frustum of a cone can be calculated using the following formula:

V (frac{1}{3}pi [R^2H - r^2h])

Where:

R is the radius of the larger base of the frustum.

r is the radius of the smaller base of the frustum.

H is the height of the entire cone from which the frustum was cut.

h is the height of the smaller cone that was removed.

Step-by-Step Derivation

Let's derive the formula step by step, starting with the volumes of the larger and smaller cones.

Volume of the Larger Cone

The volume of the larger cone is:

Vlarge (frac{1}{3}pi R^2H)

Volume of the Smaller Cone

The volume of the smaller cone that was removed is:

Vsmall (frac{1}{3}pi r^2h)

Volume of the Frustum

To find the volume of the frustum, we subtract the volume of the smaller cone from the volume of the larger cone:

Vfrustum Vlarge - Vsmall

Vfrustum (frac{1}{3}pi R^2H - frac{1}{3}pi r^2h)

Vfrustum (frac{1}{3}pi [R^2H - r^2h])

Note that (H - h) represents the height of the frustum.

Using Similar Triangles

To further illustrate the concept, let's use similar triangles to solve for (H).

Consider the following relationships from similar triangles:

(frac{H - h}{r} frac{H}{R})

From this, we can solve for (H):

H (frac{H - h}{r} times R)

H (frac{HR - hR}{r})

H (frac{Hr}{R - r})

Now, substituting (H) back into the volume formula:

Vfrustum (frac{1}{3}pi left(frac{Hr}{R - r}right)^2H - frac{1}{3}pi r^2h)

Vfrustum (frac{1}{3}pi left(frac{H^2r^2}{(R - r)^2}right) - frac{1}{3}pi r^2h)

Vfrustum (frac{1}{3}pi left(frac{H^2r^2 - r^2h(R - r)^2}{(R - r)^2}right))

Vfrustum (frac{1}{3}pi r^2 left(frac{H^2 - h(R - r)^2}{(R - r)^2}right))

Using the identity (a^3 - b^3 (a - b)(a^2 ab b^2)), we can simplify further:

Vfrustum (frac{1}{3}pi Hr^2 left(frac{R - r}{R^2 - r^2}right))

Vfrustum (frac{1}{3}pi Hr^2 left(frac{1}{R - r}right))

Vfrustum (frac{1}{3}pi hR^2r^2 left(frac{1}{R - r}right))

Conclusion

In summary, the volume of the frustum of a cone can be accurately calculated using the formula (V frac{1}{3}pi [R^2H - r^2h]). By understanding the relationships between the different components of the frustum, we can derive this formula through geometric properties and algebraic manipulation.

Key Points

The volume of a frustum of a cone is derived using the difference in volume between the larger and smaller cone. Similar triangles play a critical role in solving for the height of the frustum. The final formula is (frac{1}{3}pi [R^2H - r^2h]).

Resources

For more information on the frustum of a cone and related geometric concepts, you may find the following resources helpful:

Math is Fun: Cone Brilliant: Cone Volume Khan Academy: Volume of Cones and Cylinders