Q 62. - Calculus

Question

Prove that the lateral surface area of a right circular cone

is equal to πrl, where r is the radius of the cone and

l is the length of the diagonal of the cone, that is, the

distance from the vertex of the cone to a point on its

circumference.

Step-by-Step Solution

Answer

Hence it is proved that the lateral surface area of a cone is$\pi rl$ square units.

Step 1. Given Information

A cone's surface area is equal to $\mathrm{\pi rl}$, where $r$ is the cone's radius and $l$ is its slant height.

Step 2. Show that the lateral surface area of a right circular cone is equal to πrl

Let's consider,

r is the radius of the cone.

$l$ is the slant height.

The circumference of the base is $2\mathrm{\pi r}$

Length of $arc AB$ is $2\mathrm{\pi r}$

$\frac{\text{ Area of sector }OAB}{\text{ Area of circle with centre at }C}=\frac{\text{ Arc length }AB\text{ of sector }OAB}{\text{ Circumference of circle with centre at }O}$

$\therefore \frac{\text{ Area of sector }\mathrm{OAB}}{\pi l^2}=\frac{2\pi r}{2\pi l}=\frac{r}{l}$

So, Area of sector $OAB$,

$$\begin{gathered} A=\frac{r}{l}\times \pi l^2 \\ =\pi rl \end{gathered}$$