Question

If h be the height and $\alpha$ the semi - vertical angle of a right circular cone, then its volume is given by-

• A. $\frac{1}{3}\pi h^{3}ta{n}^2\alpha$
• B. $\frac{1}{3}\pi h^{2}ta{n}^2\alpha$
• C. $\frac{1}{3}\pi h^{2}ta{n}^3\alpha$
• D. $\frac{1}{3}\pi h^{3}ta{n}^3\alpha$

Answer 1

The correct option is A $\frac{1}{3}\pi h^{3}ta{n}^2\alpha$

Let ${}^{\prime }{r}^{\prime }$ be the radius of the base of cone.

Now, in $△OAB$,

$\tan \alpha =\frac{r}{h}\Rightarrow r=h\tan \alpha$

Now, Volume of cone $=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi {\left(h\tan \alpha \right)}^2.h=\frac{1}{3}\pi h^3{tan}^2\alpha$