Question

Consider a right angled cone with radius $R$ and height $h$, what is the volume of cone?

• A. $\frac{1}{3}\pi R^{2}h$
• B. $\frac{2}{3}\pi R^{2}h$
  
• C. $\pi R^{2}h$
  
• D. None of the above

Answer 1

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

Consider the following cone of semivertical angle $\theta$, base radius $R$, and height $H$.

Consider a small disk of radius $r$ and at a distance of $x$ from the top.

Now $\tan \theta =\frac{R}{h}=\frac{r}{x}$

Therefore, $x=\frac{r}{R}.h$

$\Rightarrow dx=\frac{dr}{R}.h$

Now the area of the disk is $\pi r^2$.

Therefore volume will be

$={\int }_{x=0}^{x=h}\pi r^2.dx$

$={\int }_{r=0}^{r=R}\pi r^2.\frac{dr}{R}.h$

$=\pi \frac{R^3.h}{3R}$

$=\frac{\pi R^{2}h}{3}$