Volume of a Frustum of a Cone

Frustum of a Cone:

When a plane parallel to the base of cone and perpendicular to its height is dividing the cone in two parts as shown in figure below then the part of the cone containing base of the cone is the frustrum of the cone.

And the volume occupied by the frustrum of cone is called as volume of the frustrum of the cone.

We know that,

Volume of cone = 1/3*π*r2*h

Derivation for Volume of frustrum of cone:

  • Let us consider the cone as shown in figure below and we have cut it by a plane parallel to its base in such way that two parts are formed. The part with base is the frustrum of cone and the part which does not contain the base is the small cone again as shown in figure.
  • Let us consider h be the height of frustrum of cone and a be the height of small cone formed.
  • Then the height of big or whole cone will be (a + h).
  • Let us consider r1 be the radius base of big cone and r2 be the radius of base of small cone.

Now, we can find the volume of frustrum of cone as,

Volume of frustrum of cone = volume of big cone – volume of small cone

As we know that volume of cone = 1/3*π*r2h

Thus, here volume of big cone = 1/3*π*r12*(a + h)

And volume of small cone = 1/3*π*r22*a

Hence, Volume frustrum of cone = 1/3*π*r12*(a + h) – 1/3*π*r22*a

= 1/3*π*[ r12*a + r12*h – r22*a]

= 1/3*π*[r12h + r12*a – r22*a]

Now, we have to find the value of a:

To find the value of a see in the following figure which represents the cone in the form of triangle.

We know that, in similar triangles the ratio of sides remains constant.

From fig. 2, the triangles ΔAEF and ΔAGB are similar triangles.

So, we can write as,

(a + h)/ r1 = a/r2

(a + h) *r2 = a*r1

Thus, a*r2 + h*r2 = a*r1

Hence, a*(r1 – r2) = h*r2

Finally, we get as, a = h*r2/ (r1 – r2)

By putting the value of a we get,

Volume of frustrum of cone = 1/3*π*[ r12*h + r12*h*r2/(r1 – r2) – r22*h*r2/(r1 – r2)]

= 1/3*π*h*[r12 + r12*r2/ (r1 – r2) – r23/ (r1 – r2)]

Hence proved.

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