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

The magic of geometry often lies in its ability to bring clarity to the complex and order to the irregular. Consider the truncated cone - a cone that's had its tip sliced off. This seemingly whimsical shape, often compared to a circus tent or lampshade, is rich in mathematical significance. How do we gauge the volume of such an object? In this guide, we will demystify the mathematics behind the truncated cone and lead you step-by-step through the process of determining its volume. Dive in as we unravel the world of frustums!

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?

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

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