The radius of a sector of a circle is 12 cm and its angle is 120°. Its straight edges are joined together to form a cone. Find the volume of this cone.

The sector is as shown in the figure with TU = 12 as radius and ∠VTU = 120°

Now edges VT and UT are joined to form the cone as shown and hence points U and V become the same

For the cone formed TU or TV is the slant height of cone

TU = l = 12 cm

To find volume of cone we need to find base radius of cone and height of cone

Radius of base of cone as seen in figure is OV or OU let us assume that to be r_c

To find base radius of cone we can use the fact that the cone is made by folding the sector which means the curved surface area of cone will be same as area of sector thus we will find those areas and equate them

Area of sector =

r is the radius of sector which is TU = 12 cm and θ is the angle which is ∠VTU = 120°

⇒ area of sector =

=

=

= 48π

Curved surface area of cone = πr_cl

= π × r_c × 12

= 12πr_c

Therefore,

⇒ 48π = 12πr_c

⇒ r_c = 4 cm

Now we have found the base radius of cone we need to find one more parameter which is the height to find volume

Let the height of the cone be h which is TO in the diagram

using the formula

l =

⇒ 12 =

Squaring both the sides

⇒ 144 = 16 + h²

⇒ h² = 144 – 16

⇒ h² = 128

⇒ h² = 64 × 2

⇒ h = 8√2 cm

Volume of cone = πr_c²h

= × × 4² × 8√2

= × × 16 × 8√2

=

=

= 134.095 × √2

= 134.095 × 1.414

= 189.61 cm³

Therefore, volume of cone is 189.61 cm³