Surface area of a coneThe 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: 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 whose base is A, 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 replaning 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 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}) 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} 




