Question

THE RADIUS OF THE BASE OF A RIGHT CIRCULAR CONE OF SEMI VERTICAL ANGLE $\alpha$ IS "$r$".SHOW THAT THE VOLUME IS $1/3$$\pi$ $r ^{3} cot \alpha$ AND CURVED SURFACE AREA IS $\pi r ^{2} cosec \alpha$

Answer 1

Imagine opening the curved surface of a cone for calculating csa

Answer 2

Proved below.

Step-by-step explanation:

Given:

Here r is the radius of the base of a right circular cone of semi vertical; angle α.

To prove:

Volume = $\frac{1}{3}\pi r&3 \cot \alpha$  

Curved surface area = $\pi r^2 cosec \alpha$

Proof:

As shown in the figure below,

$sin \alpha = \frac{r}{l}$

⇒ r sinα = l

Also, $tan \alpha =\frac{r}{h}$

$\frac{1}{cot \alpha}= \frac{r}{h}$

⇒ h = r cot α

Volume = $\frac{1}{3}\pi r^2h$

             = $\frac{1}{3} \pi r^2 rcot\alpha$

             = $\frac{1}{3} \pi r^3cot\alpha$

Surface area = $\pi rl$

                      = $\pi r \cdot rcosec\alpha$

                      =  $\pi r^2cosec\alpha$

Hence proved.