Question

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

Answer 1

Hint: The formula for the volume of a cone of radius r and height h is given by-
$\mathrm V=\dfrac13\mathrm{πr}^2\mathrm h$

Complete step-by-step solution -

Let ABC be the given sector. When we convert this sector into a cone, a few transformations will take place. The length of the arc of the sector will become the diameter of the cone. Also, the radius of the sector will become the slant height of the cone.
Radius of the sector = slant height of the cone = 12 cm = l
Length of the arc =
$=\left(\dfrac{120}{360}\right)2\mathrm\pi\left(12\right)\\=\dfrac13\times2\mathrm\pi\times12\\=8\mathrm\pi\;\mathrm{cm}$
This is equal to the circumference of the cone with radius r’.
$2\mathrm{πr}'=8\mathrm\pi\\\mathrm r'=4\;\mathrm{cm}$

Now we have to find the height of the cone using the formula-
$\mathrm r'^2+\mathrm h^2=\mathrm l^2\\4^2+\mathrm h^2=12^2\\\mathrm h^2=144-16=128\\\mathrm h=11.31\;\mathrm{cm}$

Now, we can easily find the volume of the cone-
$=\dfrac13\times{(3.14)}\times{{4}^2}\times{11.31}\\=189.5\;\mathrm{cm}^3$
This is the required answer.

Note: In this question, we have to carefully analyze what transformation will take place while converting the sector into the cone. Once we do that, apply the required conditions. Then the problem can be easily solved. Also, remember to write the units.