# Surface Area of a Right Cone

How to find the surface area of a right cone: definition and its parts, formula, proof, example, and its solution.

## Cone

This shape is called a cone.

A cone has a base (a circle) and a vertex.

The lateral face of a cone is the non-base face.

## Right Cone

A right cone is a cone

whose height meets at the center of the base circle.

The slant height is the distance between

the vertex and the endpoint of the base circle.

## Formula

*A* = *πr*^{2} + *πrh _{s}*

*A*: surface area of a right cone

*r*: radius of the base

*h*: slant height of the right cone

_{s}*πr*

^{2}: base area

*πrh*: lateral area

_{s}## Proof

Draw the net of a right cone.

(lateral area's arc) = (base circle's circumference)

2*πh _{s}*⋅(m∠[blue] / 360) = 2

*πr*

Then, after cancelling 2*π*,

m∠[blue] / 360 = *r* / *h _{s}*.

(base area) = *πr*^{2}

(lateral area) = *πh _{s}*

^{2}⋅(m∠[blue]/360)

Area of a circular sector

*A*=

*πr*

^{2}+

*πh*

_{s}^{2}⋅(m∠[blue]/360)

m∠[blue]/360 = *r*/*h _{s}*

So

*A*=

*πr*

^{2}+

*πh*

_{s}^{2}⋅(m∠[blue]/360)

=

*πr*

^{2}+

*πh*

_{s}^{2}⋅(

*r*/

*h*)

_{s}=

*πr*

^{2}+

*πrh*.

_{s}## Example

See the right triangle in the cone.

Its legs are 3 and 4.

So it's a (3, 4, 5) triangle.

So *h _{s}* = 5.

*r* = 3, *h _{s}* = 5

*A*=

*π*⋅3

^{2}+

*π*⋅3⋅5