# Surface area of a cone

The surface area of a cone can be derived from the

surface area of a square pyramid.

Start with a square pyramid and just keep increasing the number of sides of the base. After a very large number of sides, you can see that the figure will eventually look like a cone.This is shown below:

This observation is important because we can use the formula of the surface area of a square pyramid to find that of a cone.

l is the slant height.

The area of the square is s

^{2}
The area of one triangle is (s ×

l)/2

Since there are 4 triangles, the area is 4 × (s ×

l)/2 = 2 × s ×

l
Therefore, the surface area, call it SA is:

SA = s

^{2} + 2 × s ×

l
.

Generally speaking, to find the surface area of any regular pyramid where A is the area of the base, the perimeter is P, and the slant height is

l, we use the following formula:

S = A + 1/2 (P ×

l)

Again A is the area of the base. For a figure with 4 sides, A = s

^{2} with s = length of one side.

Where does the 1/2 (P ×

l) come from?

Let s be the length of the base of a regular pyramid. Then, the area of one triangle is (s ×

l)/2

For n triangles and this also means that the base of the pyramid has n sides, we get ( n × s ×

l)/2

Now P = n × s. When n = 4, of course, P = 4 × s as already shown.

Therefore, after replacing n × s by P, we get S = A + 1/2 (P ×

l)

Let us now use this fact to derive the formula of the surface area of a cone

## How to derive the formula to get the surface area of a cone?

For a cone, the base is a circle, so A = π × r

^{2}
P = 2 × π × r

To find the slant height,

l, just use the Pythagorean Theorem

l = r

^{2} + h

^{2}
l = √ (r

^{2} + h

^{2})

Putting it all together, we get:

S = A + 1/2 (P ×

l)

S = π × r

^{2} + 1/2 ( 2 × π × r × √ (r

^{2} + h

^{2})

S = π × r

^{2} + π × r × √ (r

^{2} + h

^{2})

## 3 examples showing how to find the surface area of a cone.

**Example #1:**
Find the surface area of a cone with a radius of 4 cm, and a height of 8 cm

S = π × r

^{2} + π × r × √ (r

^{2} + h

^{2})

S = 3.14 × 4

^{2} + 3.14 × 4 × √ (4

^{2} + 8

^{2})

S = 3.14 × 16 + 12.56 × √ (16 + 64)

S = 50.24 + 12.56 × √ (80)

S = 50.24 + 12.56 × 8.94

S = 50.24 + 112.28

S = 162.52 cm

^{2}
**Example #2:**
Find the surface area of a cone with a radius of 9 cm, and a height of 12 cm

S = π × r

^{2} + π × r × √ (r

^{2} + h

^{2})

S = 3.14 × 9

^{2} + 3.14 × 9 × √ (9

^{2} + 12

^{2})

S = 3.14 × 81 + 28.26 × √ (81 + 144)

S = 254.34 + 28.26 × √ (225)

S = 254.24 + 28.26 × 15

S = 254.24 + 423.9

S = 678.14 cm

^{2}