Question

The central angles of two sectors of a circle are ${60}^o$ and ${120}^o$. Using these sectors two cones are made.
(a) If the radius of the smaller cone is $5$ centimetres, find the radius and base area of the larger cone.
(b) Find the surface area of the larger cone.

Answer 1

(a) Let $r$ be the radius of the smaller cone and $R$ be the radius of the larger cone.
Thus, $r=5cm$
Since, both the sectors are from the same circle.
Slant height of smaller cone $=$ slant height of larger cone $=l$
For smaller cone,
$\frac{r}{l}=\frac{centralangle}{{360}^o}\Rightarrow \frac{5}{l}=\frac{{60}^o}{{360}^o}$
$\Rightarrow l=\frac{{360}^o\times 5}{{60}^o}=30cm$
For larger cone,
$\frac{R}{l}=\frac{centralangle}{{360}^o}\Rightarrow \frac{R}{30}=\frac{{120}^o}{{360}^o}\Rightarrow R=\frac{{120}^o\times 30}{{360}^o}=10cm$
Base area of larger cone $=\pi R^2=\pi \times 10\times 10=100\pi {cm}^2$
(b) Surface area of the larger cone $=\pi R^2+\pi Rl=100\pi +30\times 10\times \pi =100\pi +300\pi =400\pi {cm}^2$