1. Cone
A right circular cone is a 3D solid with:
- a circular base of radius \(r\)
- a curved surface that narrows to a single point (the vertex)
- a slant height \(l\), which is the distance from the vertex to any point on the circular boundary
The slant height \(l\) is different from the vertical height \(h\). They are related by the Pythagorean theorem:
\( l = \sqrt{h^2 + r^2} \)
2. Curved Surface Area (CSA) of a Cone
The curved surface area refers to the smooth sloping surface of the cone, excluding the base.
When this curved surface is cut along one side and flattened out, it becomes a sector of a circle with radius equal to the slant height \(l\).
The formula for the curved surface area is:
\( \text{CSA} = \pi r l \)
2.1. Example
A cone has radius \(r = 5\,\text{cm}\) and slant height \(l = 13\,\text{cm}\). Its curved surface area is:
\( \text{CSA} = \pi r l = \pi \times 5 \times 13 = 65\pi \)
Using \(\pi = 3.14\):
\( 65 \times 3.14 = 204.1\,\text{cm}^2 \)
3. Total Surface Area (TSA) of a Cone
The Total Surface Area (TSA) includes:
- the curved surface area
- the area of the circular base
The base has area:
\( \pi r^2 \)
So the total surface area becomes:
\( \text{TSA} = \pi r l + \pi r^2 = \pi r (l + r) \)
3.1. Example
For a cone with radius \(r = 7\,\text{cm}\) and slant height \(l = 10\,\text{cm}\):
\( \text{TSA} = \pi r (l + r) = \pi \times 7 \times (10 + 7) = 7\pi \times 17 = 119\pi \)
Using \(\pi = 3.14\):
\( 119 \times 3.14 = 373.66\,\text{cm}^2 \)
4. Real-Life Applications of Cone Surface Area
Cones appear in many real-world scenarios, and knowing their surface areas helps in several practical tasks:
- Designing ice-cream cones (curved surface for the waffle)
- Making conical tents (cloth needed = CSA)
- Constructing cone-shaped packaging
- Calculating paint area for cone-shaped pillars
Use CSA when the base is not included, and TSA when the entire cone needs to be covered.