--- title: Surface Area of a Cone --- ## Surface Area of a Cone A cone is a three-dimensional solid that has a circular base, which is connected by a curved surface to its vertex. The curved surface of the cone is formed by a set of line segments that connect the vertex to the circumference of the circle at the bottom. The `radius of the cone 'r'` is defined by the radius of the circle formed at the bottom. The `slant height 'l'` is the distance from any point on the circle to the vertex of the cone. Lastly, `the altitude 'h'` is defined by the distance measured from the vertex to the cirle's center. * The `slant height 'h'` is calculated as: $l = \sqrt{r^2+a^2}$ * The `lateral surface area` is given by: $L = \pi r l = \pi r \sqrt{r^2+h^2}$ * The `base surface area` is given by: $B = \pi r^2$ Hence, the total surface area of the cone is the sum of the `lateral surface area` and the `base surface area`. $A= L + B = \pi r l + \pi r^2 = \pi r (l + \pi r) = \pi r (\sqrt{r^2+h^2}+ \pi r)$

#### More Information: * The derivation for the formula can be found at - [mathopenref.com](https://www.mathopenref.com/coneareaderivation.html) * The link to a video explaining the derivation is given below: