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Question

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


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Solution

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

Radius of circle, r=12cm.

Angle of sector, θ=120°

The length of an arc is given as,

l=θ360°×2πr=120°360°×2π×12=8π

Circumference, of the circle, is given by:

C=2πR

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

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

2πR=8πR=4

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

SlantHeight(l)=r=12

Also, in a cone

Slantheight2=radiusofbaseofcone2+heightofcone2l2=R2+h2122=42+h2144=16+h2144-16=h2h=128h=11.28cm

Step 2: Find the volume of cone.

Volume of cone:

V=13πR2h=13π×42×11.28=13π×16×11.28=189.4cm3

Hence, the volume of the cone is 189.4cm3.


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