Question

A sector of a circle of radius $12\mathrm{cm}$ has an angle $120°$ . By coinciding its straight edges a cone is formed. Find the volume of cone.

Answer 1

Step 1: Find the radius and height of the cone.

Radius of circle, $r=12\mathrm{cm}$.

Angle of sector, $\theta =120°$

The length of an arc is given as,

$$\begin{gathered} l=\frac{\theta }{360°}\times 2\mathrm{\pi }r \\ =\frac{120°}{360°}\times 2\mathrm{\pi }\times 12 \\ =8\mathrm{\pi } \end{gathered}$$

Circumference, of the circle, is given by:

$C=2\mathrm{\pi }R$

Where $R$ is the radius of the base of the cone formed.

Length of an arc is the circumference of the base circle. So,

$$\begin{gathered} 2\mathrm{\pi }R=8\mathrm{\pi } \\ \Rightarrow \mathrm{R}=4 \end{gathered}$$

As, slant height of the cone is equal to the radius of the sector used form cone

$\mathrm{Slant}\mathrm{Height}\left(l\right)=r=12$

Also, in a cone

$\begin{array}{rcl}\mathrm{Slant}{\mathrm{height}}^2& =& \mathrm{radius}\mathrm{of}\mathrm{base}\mathrm{of}{\mathrm{cone}}^2+\mathrm{height}\mathrm{of}{\mathrm{cone}}^2\\ l^2& =& R^2+h^2\\ & \Rightarrow & {12}^2=4^2+h^2\\ & \Rightarrow & 144=16+h^2\\ & \Rightarrow & 144-16=h^2\\ & \Rightarrow & h=\sqrt{128}\\ \therefore h& =& 11.28\mathrm{cm}\end{array}$

Step 2: Find the volume of cone.

Volume of cone:

$$\begin{gathered} V=\frac{1}{3}\mathrm{\pi }{R}^{\mathit{2}}h \\ \mathit{}\mathit{}\mathit{}=\frac{\mathit{1}}{\mathit{3}}\mathrm{\pi }\times 4^{\mathit{2}}\times 11.28 \\ =\frac{\mathit{1}}{\mathit{3}}\mathrm{\pi }\times 16\times 11.28 \\ =189.4{\mathrm{cm}}^3 \end{gathered}$$

Hence, the volume of the cone is $189.4{\mathrm{cm}}^3$.