The surface of a cone can be derived from the area of a square pyramid

Start with a pyramid square 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 be finally look like a cone.This is illustrated below:

This observation is important because we can use the formula of the surface of a pyramid square to conclude that a cone l is the height of the slope.

The area of the square is s2

The area of a triangle is (s × l) / 2

Since there are 4 triangles, the region is 4 × (s × l) / 2 = 2 × s × l

Therefore, the area, call it her is:

SA = s2 + 2 × s × l:

In General, to find the area of a regular pyramid whose base is a, the perimeter is p and the height of the slope is l, we use the following formula:

S = A + 1/2 (P × l)

Once again a is the area of the base. For a figure with 4 sides, A = s2 with s = length on one side

Where the 1/2 (P × l) come from?

Either s the length of the base of a regular pyramid. Then, the area of a 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 it has already shown.

We have, after replaning × s n by P, S = A + 1/2 (P × l)

Let us now use it to derive the formula for the surface of a cone

For a cone, the base is a circle, A = p × r2

P = 2 × p × r

To find the tilt, height, l, just use the Pythagorean theorem

l = r2 + h2

l = v (r2 + h2)

Putting all together, we get:

S = A + 1/2 (P × l)

S = p × r2 + 1/2 (2 × p × r × v (r2 + h2))

S = p × r2 + p × r × v (r2 + h2)

**Example # 1:**

Find the surface of a cone with a radius of 4 cm and 8 cm high

S = p × r2 + p × r × v (r2 + h2)

S = 3.14 × 42 + 3.14 × 4 × v (42 + 82)

S = 3.14 × 16 + 12.56 × v (16 + 64)

S = 50.24 + 12.56 × v (80)

S = 50.24 + 12.56 × 8.94

S = 50.24 + 112.28

S = cm2 162.52

**Example # 2:**

Find the surface of a cone with a radius of 9 cm and 12 cm in height

S = p × r2 + p × r × v (r2 + h2)

S = 3.14 × 92 + 3.14 × 9 × v (92 + 122)

S = 3.14 × 81 + 28.26 × v (81 + 144)

S = 254.34 + 28,26 × v (225)

S = 254.24 + 28.26 × 15

S = 254.24 + 423.9

S = cm2 678.14