Question

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

• 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 $(2\sqrt{2}-2)\pi$

TSA of cone $=\pi rl+\pi r^2=\pi r(l+r)=\pi \left[Given\right]$

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

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

CSA of cone $=\pi rl=$ CSA of sector

$\Rightarrow CSAofsector=\pi (\sqrt{2}-1)\times 2=(2\sqrt{2}-2)\pi$