How to Calculate the Volume of a Truncated Cone: Step-by-Step Guide

Step-by-step Guide: Volume of a Truncated Cone (Frustum)

The truncated cone, technically termed a ‘frustum’, can be thought of as a cone minus a smaller cone removed from its top. To find its volume, we can utilize the formula for the volume of a cone and modify it accordingly.

Volume of a Frustum:
\( V = \frac{1}{3}\pi h(R^2 + r^2 + Rr) \)

Where:

• \(( R ) =\) radius of the larger base
• \(( r ) =\) radius of the smaller base (the section that was removed)
• \(( h ) =\) height of the frustum (not the original cone’s height)

Examples

Example 1:
Find the volume of a frustum with \( R = 8 \text{ cm}\), \( r = 4 \text{ cm}\), and \( h = 10 \text{ cm}\).

Solution:
Using the formula:
\( V = \frac{1}{3}\pi(10)(8^2 + 4^2 + 8 \times 4) \)
\( V \approx \frac{1}{3}\pi(10)(64 + 16 + 32) \)
\( V \approx \frac{1}{3}\pi(10)(112) \)
\( V \approx 1172.26 \text{ cm}^3 \)

Practice Questions:

1. What is the volume of a frustum with \( R = 10 \text{ cm}\), \( r = 6 \text{ cm}\), and \( h = 12 \text{ cm}\)?
2. A truncated cone has \( R = 15 \text{ cm}\), \( r = 5 \text{ cm}\), and \( h = 20 \text{ cm}\). What is its volume?

Answers:

1. \( \approx 2461.76 \text{ cm}^3 \)
2. \( \approx 6803.3 \text{ cm}^3 \)