# Surface Area of a Cone

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The **surface area of a cone** (if the area of the base is included) is \(\pi r^2 + \pi L r \), where \(r\) denotes the radius of the base of the cone, and \(L\) denotes the slant height of the cone. The surface area of a cone without the base area is sometimes called the lateral area.

**Note:** This wiki talks about *right* circular cones.

#### Contents

## Derivation of the Formula

In the cone below, \(h\) is the height, \(L\) is the slant height, \(c\) is the circumference of the base, and \(r\) is the radius of the base.. How about if we unroll it?

If we unroll it, the shape is

It is a sector of a circle with radius \(L\) and arc length \(c\). So the surface area of the cone excluding the area of the base is the area of the sector above. The area of a sector given the arc length (\(c)\) and radius \(L\) is given by \(A=\dfrac{1}{2}cL\). Now applying this to the cone, we have\(A=\dfrac{1}{2}cL\) where \(L\) is the slant height and \(c\) is the circumference of the base. After some manipulations, \(A=\pi Lr\) as given in the definition.

## Examples

## A cone was formed by rolling a thin shit of metal in the form of a sector of a circle \(72\) cm in diameter with central angle of \(150^\circ\). Find the surface area (excluding the area of the base) of the cone formed.

The surface area is the area of the sector of a circle, so we have,

\(A=\dfrac{\theta}{360}(area~of~the~circle)=\dfrac{150}{360}(\pi)(36^2)=540 \pi\) \(_\square\)

A pile of sand is in the shape of a right circular cone. It has a height of \(3\) feet and a base diameter of \(8\) feet. Find it's surface area including the area of the base.

After some manipulations, we can see that a \(3-4-5\) right triangle is formed. Therefore, the slant height is \(5\). So,

\(A=\pi r^2+\pi L r=\pi (4^2) +\pi(5)(4)=36 \pi\) \(_\square\)

What is the lateral area of a cone that has a height of \(8\) and a radius of \(5\)?

Note that lateral area is the surface area excluding the area of the base.

First step is to compute for the slant height, \(L\). Applying pythagorean theorem, we get \(\sqrt{89}\). Applying the formula, we have

\(A= \pi Lr=\pi (\sqrt{89})(5)=5 \pi \sqrt{89}\) \(_\square\)

**Cite as:**Surface Area of a Cone.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/surface-area-of-a-cone/