1. Understanding a Frustum
A frustum of a cone is formed when the top of a cone is cut off by a plane parallel to its base. It has:
- a larger circular base of radius \(R\)
- a smaller circular top of radius \(r\)
- a slant height \(l\), which is the distance along the sloping surface
Frustums appear in real life as buckets, lampshades, flower pots, and cones with the top removed.
2. Curved (Lateral) Surface Area of a Frustum
The curved surface area (CSA) includes only the outer sloping surface between the two circular edges.
For a frustum with radii \(R\) and \(r\), and slant height \(l\):
\( \text{CSA} = \pi (R + r) l \)
This formula makes sense because the curved surface opens up into a trapezium-shaped figure whose parallel sides are \(R\) and \(r\) and whose height is the slant height \(l\).
2.1. Example
A frustum has radii \(R = 10\,\text{cm}\), \(r = 6\,\text{cm}\), and slant height \(l = 8\,\text{cm}\).
\( \text{CSA} = \pi (R + r) l = \pi (10 + 6) \times 8 = 16\pi \times 8 = 128\pi \)
Using \(\pi = 3.14\):
\( 128 \times 3.14 = 401.92\,\text{cm}^2 \)
3. Total Surface Area of a Frustum
The total surface area (TSA) includes:
- the curved surface area
- the area of the larger base = \(\pi R^2\)
- the area of the smaller top = \(\pi r^2\)
So the total surface area is:
\( \text{TSA} = \pi (R + r) l + \pi R^2 + \pi r^2 \)
3.1. Example
For a frustum with \(R = 7\,\text{cm}\), \(r = 3\,\text{cm}\), and \(l = 10\,\text{cm}\):
\( \text{TSA} = \pi (R + r) l + \pi R^2 + \pi r^2 \)
\( = \pi (7 + 3) \times 10 + \pi (49) + \pi (9) \)
\( = 100\pi + 49\pi + 9\pi = 158\pi \)
Using \(\pi = 3.14\):
\( 158 \times 3.14 = 495.\,\text{cm}^2 \)
4. Real-Life Applications
Frustums are used widely in practical designs. Knowing their surface area helps in:
- Calculating metal sheets needed to make buckets or pots
- Finding cloth required for lampshades
- Painting or polishing decorative frustum-shaped pillars
- Packaging design for frustum-shaped containers
Use CSA when only the sloping surface matters, and TSA when the entire object is exposed.