1. Understanding a Frustum
A frustum of a cone is formed when a cone is cut by a plane parallel to its base, removing the top portion. It has:
- a larger circular base of radius \(R\)
- a smaller circular top of radius \(r\)
- a height \(h\) (perpendicular distance between the two bases)
Frustums appear in objects like buckets, lampshades, flower pots, glasses, and truncated cones.
2. Formula for Volume of a Frustum
The frustum is essentially the difference between the volume of a large cone and a smaller cone removed from its top.
The formula for the volume is:
\( \text{Volume} = \dfrac{1}{3} \pi h (R^2 + r^2 + Rr) \)
This expression works because the frustum has circular cross-sections that change smoothly from \(R\) to \(r\).
2.1. Example
A frustum has height \(h = 12\,\text{cm}\), larger radius \(R = 8\,\text{cm}\), and smaller radius \(r = 4\,\text{cm}\).
Volume:
\( \text{Volume} = \dfrac{1}{3} \pi h (R^2 + r^2 + Rr) \)
\( = \dfrac{1}{3} \pi (12)(64 + 16 + 32) \)
\( = \dfrac{1}{3} \pi (12)(112) = 448\pi \)
Using \(\pi = 3.14\):
\( 448 \times 3.14 = 1406.72\,\text{cm}^3 \)
This is the internal capacity of the frustum.
3. Why the Formula Makes Sense (Simple Intuition)
If you imagine the frustum as a "cone with its top cut off", then the volume remaining depends on:
- area of the two circular bases
- smooth change of radius from \(R\) to \(r\)
- the height \(h\)
The expression \(R^2 + r^2 + Rr\) comes from averaging the areas of the circular cross-sections along the height.
4. Real-Life Applications
Volume of frustums is frequently used in:
- Manufacturing buckets, mugs, and containers
- Designing lampshades
- Calculating material for frustum-shaped pillars
- Estimating capacity of decorative pots
Whenever the object is shaped like a cut cone, use \(\dfrac{1}{3} \pi h (R^2 + r^2 + Rr)\) to find its capacity.