Volume of frustum of cone

What is Frustum of Cone?

Frustum of Cone: The frustum of a cone is the portion of the cone that is cut off by the planes perpendicular to the cone’s axis.

The Volume of Frustum of Cone

When a cone is cut by a plane parallel to its base, a frustum is formed. The volume of the frustum is the volume of the cone minus the volume of the cone’s two bases.

To calculate the volume of a frustum, we need to know the height, h, of the frustum, the radius of the base, r, and the slant height, s. The volume of a cone is given by:

V = 1/3 * pi * r^2 * h

The volume of the frustum is given by:

V = 1/3 * pi * r^2 * h – (1/3 * pi * r^2 * h)

The volume of a frustum of a cone depends on its slant height and radius of the upper and bottom circular part. Basically a frustum of a cone is formed when we cut a right-circular cone by a plane parallel to its base into two parts. Hence, this part of the cone has its surface area and volume.

Let us learn here to derive the volume of frustum and understand the concept better by solving the problems.

Derivation of Volume of Frustum of Cone

Let us consider a right circular cone which cut by a plane parallel to its base as given in the below figure.

Here, we can consider the frustum as the difference between the two right circular cones.

Let,

Height of large cone = h

Slant height of large cone = l

and radius = r

Now,

Height of smaller cone = h’

Slant height of smaller cone = l’

And radius = r’

In case of frustum,

Height of frustum = H

Slant height= L

Now, we can say,

volume of bigger cone say V₁ = ⅓ π r²h

Also, volume of smaller cone, V₂ = ⅓ πr’²h’

Therefore,

Volume of frustum of cone, V = V₁ – V₂

V = ⅓ π r²h – ⅓ πr’²h’

V = ⅓ π (r²h – r’²h’) ……………(1)

From the figure, in the ∆OO’D and ∆OPB;

∠DOO’ = ∠BOP (Common Angle)

CD//AB (Plane cutting the cone is parallel to the base)

⇒∠O’DO = ∠PBO (Corresponding Angles)

Thus, ∆OO’D~∆OPD (By AA criterion of similarity)

As we know, by the condition of similarity, the ratio of sides of similar triangles are equal.

Formula to Calculate the Volume of Frustum of Cone

The volume of the frustum of a cone given by the following formula:

Volume = (1/3) * pi * h * (r1^2 + r2^2)