Question

A cone is made from a circular sector of radius $2$ units. If the total surface area of cone (including base) is $\pi$ sq units, find the area of the sector.

• A. $(3-2\sqrt{2})\pi$
• B. $(2\sqrt{2}-2)\pi$
• C. $\sqrt{2}\pi$
• D. $\sqrt{3}\pi$

Answer 1

The correct option is B $(2\sqrt{2}-2)\pi$

Given, circular sector of radius $2cm$, i.e. slant height of the cone, $l=2$.

Also, Total Surface Area of cone $=\pi r(l+r)$

$\Rightarrow \pi =\pi r(l+r)\Rightarrow r(l+r)=1$

$\Rightarrow r(2+r)=1\Rightarrow 2r+r^2=1\Rightarrow r^2+2r+1=2$

$\Rightarrow {(r+1)}^2=2\Rightarrow r=(\sqrt{2}-1)$.

Here, Curved Surface Area of cone $=$ Curved Surface Area of sector $=\pi rl=\pi (\sqrt{2}-1)\times 2$

$\therefore$ Surface Area of sector $=2\pi (\sqrt{2}-1)=\pi (2\sqrt{2}-2)$.

Hence, option $B$ is correct.