# Surface Area of a Cone

The
**
total
**
surface area
of a
cone
is the sum of the area of its base and the lateral (side) surface.

The
**
lateral surface area
**
of a cone is the area of the lateral or side surface only.

Since a cone is closely related to a pyramid , the formulas for their surface areas are related.

Remember, the formulas for the lateral surface area of a pyramid is $\frac{1}{2}pl$ and the total surface area is $\frac{1}{2}pl+B$ .

Since the base of a cone is a circle, we substitute $2\pi r$ for $p$ and $\pi {r}^{2}$ for $B$ where $r$ is the radius of the base of the cylinder.

So, the formula for the
**
lateral surface area
**
of a right cone is
$L.S.A=\pi rl$
, where
$l$
is the slant height of the cone
*
.
*

**
Example 1:
**

Find the lateral surface area of a right cone if the radius is $4$ cm and the slant height is $5$ cm.

$L.S.A=\pi \left(4\right)\left(5\right)=20\pi \approx 62.82\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\text{cm}}^{2}$

The formula for the
**
total surface area
**
of a right cone is
$T.S.A=\pi rl+\pi {r}^{2}$
.

**
Example 2:
**

Find the total surface area of a right cone if the radius is $6$ inches and the slant height is $10$ inches.

$\begin{array}{l}T.S.A=\pi \left(6\right)\left(10\right)+\pi {\left(6\right)}^{2}\\ =60\pi +36\pi \\ =96\pi \text{\hspace{0.17em}}{\text{inches}}^{2}\\ \approx 301.59\text{\hspace{0.17em}}{\text{inches}}^{2}\end{array}$