Volume of Frustum of Cone

A frustum of a cone is the solid obtained when the top portion of the cone is cut off by a plane parallel to its base. This solid shape is seen even in our real life, such as in buckets, glasses, containers, lampshades, etc. Understanding how to calculate the volume of a frustum is an important geometrical skill that has practical applications in engineering, architecture, and design. In this guide, we will learn the formula for the volume of a frustum of a cone, explore its step-by-step derivation, and solve examples to build a strong understanding of the concept.

Table of Contents

• What is the Volume of a Frustum of a Cone
• Derivation of Volume of a Frustum of a Cone
• Types and Properties of the Volume of a Frustum of a Cone
• Solved Examples on Volume of a Frustum of a Cone

What is the Volume of a Frustum of a Cone

A frustum of a cone is the solid formed when a plane parallel to the base cuts through a cone, removing its top portion.

Volume of frustum of a cone, V = (πh/3)(R² + Rr + r²),
where

• R = radius of the larger base (bottom)

• r = radius of the smaller top (upper face)

• h = perpendicular height (vertical distance between the two parallel faces)

• l = slant height = √[h² + (R − r)²]
  
  

Derivation of Volume of a Frustum of a Cone

Consider a full cone with height H and base radius R. A smaller cone of height (H − h) and radius r is cut from the top.

Let H and h be the heights of the cone and frustum, respectively. Let L and l be the slant heights of the cone and the frustum, respectively. Let R and r be the radii of the circular bases.

Triangles ABC and ADE are similar. Hence, the ratio of the corresponding sides is proportional, r/R = (H − h)/H

so H = Rh/(R − r).

Volume of full cone = (1/3)πR²H = (1/3)πR² × Rh/(R − r) = πR³h / [3(R − r)].

Height of small cone = H − h = Rh/(R − r) − h = rh/(R − r).

Volume of small cone = (1/3)πr² × rh/(R − r) = πr³h / [3(R − r)].

Volume of frustum = Volume of full cone − Volume of small cone.

= πh / [3(R − r)] × (R³ − r³)

Using the identity: R³ − r³ = (R − r)(R² + r² + Rr)

= πh/3 × (R² + r² + Rr)

= (πh/3)(R² + r² + Rr)

Types and Properties of the Volume of a Frustum of a Cone

CONDITION	SHAPE	VOLUME
r = 0	Full cone	(1/3)πR²h
R = r	Cylinder	πR²h
R > r > 0	Frustum	(πh/3)(R² + r² + Rr)

 

Solved Examples on Volume of a Frustum of a Cone

Example 1: Find the volume of a frustum with R = 14 cm, r = 7 cm, and h = 6 cm. (Use π = 22/7)
Solution: Given: R = 14, r = 7, h = 6
Using V = (πh/3)(R² + r² + Rr)
R² = 196, r² = 49, Rr = 98
R² + r² + Rr = 196 + 49 + 98 = 343
V = (22/7)(6/3)(343) = (22/7)(2)(343)
= (22/7) × 686
= 22 × 98
= 2156 cm³
∴ Volume of frustum is 2156 cm³.

Example 2: A bucket is in the shape of a frustum with top radius 28 cm, bottom radius 21 cm, and height 30 cm. Find its capacity in litres. (Use π = 22/7)
Solution: R = 28 cm, r = 21 cm, h = 30 cm
Substituting values in the equation V = (πh/3)(R² + r² + Rr)
V = (π 30/3)(28² + 21² + 28 × 21)
= 399,060/7
= 57,008.57 cm³
1 litre = 1000 cm³
∴ V = 57,008.57/1000 = 57.01 litres.
∴ the capacity of the bucket is 57.01 litres.

Example 3: Find the volume of a frustum with R = 6 cm, r = 3 cm, h = 4 cm.
Solution: Given R = 6 cm, r = 3 cm, h = 4 cm
V = (πh/3)(R² + r² + Rr)
V = (π × 4/3)(63)
V = (4π/3)(63)
V = 84π cm³
∴ Volume of frustum is 84 cm³.