# A cone has a perpendicular height of 12cm and slant height of 13cm. Calculate it's total surface area (take pie r =3.142)?

Apr 16, 2018

The surface area is 282.78 $c {m}^{2}$

#### Explanation:

The perpendicular height, $h$, and the radius, $r$, of the base of the cone form the legs of a right triangle with the slant height, $l$, as the hypotenuse of that right triangle. So we can use the Pythagorean Theorem to determine the radius of the base of the cone in terms of the perpendicular height and the slant height.

Equation I
${r}^{2} = {l}^{2} - {h}^{2}$

Equation II
$r = \sqrt{{l}^{2} - {h}^{2}}$

The total surface area of a cone, $A$ is the cone's lateral surface area, ${A}_{l}$ plus the area of the base, ${A}_{b}$.

Equation III
$A = {A}_{l} + {A}_{b}$

The formula for the lateral surface area is

Equation IV
${A}_{l} = \pi l r$.

The formula for the area of the base is

Equation V
${A}_{b} = \pi {r}^{2}$

Substitute Equations IV and V into Equation III.

Equation VII
$A = \pi l r + \pi {r}^{2}$

Substitute Equations I and II into Equation VII.

$A = \pi l \sqrt{{l}^{2} - {h}^{2}} + \pi \left({l}^{2} - {h}^{2}\right)$

Now evaluate with $l = 13$, $h = 12$ and $\pi = 3.142$.

$A = 3.142 \cdot 13 \sqrt{{13}^{2} - {12}^{2}} + 3.142 \left({13}^{2} - {12}^{2}\right)$

$A = 3.142 \cdot 13 \cdot 5 + 3.142 \cdot 25 = 282.78$ $c {m}^{2}$