Question

The semivertical angle of a cone is $45^{\circ }$. If the height of the cone is $20.025cm$, then the approximate value of its lateral surface area (in sq. $cm$ ) is

• A. $401\sqrt{2}\pi$
• B. $400\sqrt{2}\pi$
• C. $402\sqrt{2}\pi$
• D. $405\sqrt{2}\pi$

Answer 1

Answer: (A)

Given, semi vertical angle $\theta =45^{\circ }$

In $\Delta AOB$,
$\tan 45^{\circ }=\frac{r}{h}$
$1=\frac{r}{h}$
$\Rightarrow r=h$
and $l^2=r^2+h^2$
$l=\sqrt{h^2+h^2}[\because r=h]$
$l=\sqrt{2h^2}=\sqrt{2}h$
Lateral surface area
$(S)=\pi rl$
$=\pi h\cdot \sqrt{2}h$
$S=\sqrt{2}\pi h^2[\because h=20.025cm]$
$=\sqrt{2}\pi (20.025{)}^2$
$=\sqrt{2}\pi \times 401$
$S=401\sqrt{2}\pi$