Volume of Frustum with Two Radii and Height Examples

Volume of Frustum is calculated for the portion of a cone left between two parallel circular bases. This page covers the volume of frustum formula, meanings of the radii and height, unit conversion, solved examples, and quick methods for aptitude problems.

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What Is the Volume of a Frustum?

The volume of a frustum is the space occupied by a cone after its top portion is removed by a cut parallel to the base. For larger radius R, smaller radius r, and perpendicular height h, its volume is V = (1/3)πh(R² + Rr + r²).

Here, R and r are the radii of the larger and smaller circular bases, respectively, and h is the perpendicular distance between the bases. The formula uses both base radii and not the slant height. If the frustum is formed by cutting a cone, its volume can also be found by subtracting the volume of the smaller cone from the volume of the original cone.

Volume of Frustum Video Lessons

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Frustum Volume and Lateral Surface Area

Learn how to calculate the volume and lateral surface area of a frustum using the relevant mensuration formulas and dimensions.

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Volume of Frustum Formula & Tricks

Important Formulas

Volume of a frustum
V = (1/3)πh(R² + Rr + r²)

R is the larger radius, r is the smaller radius, and h is the perpendicular height between the circular bases.

Volume using diameters
V = (πh/12)(D² + Dd + d²)

D and d are the larger and smaller diameters. This follows from substituting R = D/2 and r = d/2.

Frustum formed from a cone
V = (1/3)π(R²H - r²x)

H is the height of the original cone and x is the height of the removed smaller cone. Both cones have the same vertex and axis.

Relation between cone heights
x/H = r/R

For similar cones, the height ratio equals the corresponding radius ratio. Therefore, x = Hr/R.

Slant height relation
l² = h² + (R - r)²

The slant height l, perpendicular height h, and difference of radii R - r form a right triangle.

Quick Tricks

Use the three-term radius expression

Calculate R² + Rr + r² first, then multiply by πh/3. This reduces errors when the radii are small integers.

Example: For R = 5 and r = 3, R² + Rr + r² = 25 + 15 + 9 = 49.
Use diameters directly when given

When diameters are provided, use V = (πh/12)(D² + Dd + d²) instead of converting both diameters into radii.

Example: For D = 10, d = 6, and h = 7, the bracket is 100 + 60 + 36 = 196.
Check the limiting cases

If r = 0, the frustum becomes a cone and the formula becomes V = (1/3)πR²h. If R = r, it becomes a cylinder and the formula becomes V = πR²h.

Example: With R = r = 4 and h = 6, V = π × 4² × 6 = 96π cubic units.

Volume of Frustum Concepts

Applying the Frustum Volume Formula

Substitute the larger radius, smaller radius, and perpendicular height into V = (1/3)πh(R² + Rr + r²).

Keep all lengths in the same unit before substitution. For R = 5 cm, r = 3 cm, and h = 4 cm, V = (1/3)π × 4 × (25 + 15 + 9) = 196π/3 cm³.

Example: Using π = 22/7, the volume is approximately 205.33 cm³.

Volume from Diameters

If the two circular bases have diameters D and d, use V = (πh/12)(D² + Dd + d²).

This form avoids separately dividing each diameter by 2. For D = 10 m, d = 6 m, and h = 7 m, V = (π × 7/12)(100 + 60 + 36) = (343π/3) m³.

Example: Using π = 22/7, V = 358.67 m³ approximately.

Frustum as the Difference of Two Cones

A frustum can be treated as the original cone minus the smaller cone removed from its top.

If the original cone has radius R and height H, and the removed cone has radius r and height x, then V = (1/3)π(R²H - r²x). Since the cones are similar, x/H = r/R.

Example: If R = 6 cm, r = 3 cm, and H = 12 cm, then x = 12 × 3/6 = 6 cm. The frustum volume is (1/3)π(36 × 12 - 9 × 6) = 126π cm³.

Perpendicular Height and Slant Height

The volume formula requires perpendicular height h, not slant height l.

If slant height and radii are given, first calculate h using h = √(l² - (R - r)²). For l = 13 cm, R = 8 cm, and r = 3 cm, h = √(169 - 25) = 12 cm.

Example: The volume is then calculated with h = 12 cm in the frustum formula.

Units and Dimensional Check

Volume must be expressed in cubic units because the formula multiplies height by a squared-length expression.

If all dimensions are in centimetres, the answer is in cm³. Common conversions are 1 m³ = 1,000,000 cm³ and 1 litre = 1,000 cm³. Convert before calculating when the dimensions use different units.

Example: A volume of 2.5 m³ equals 2,500 litres because 2.5 × 1,000 = 2,500.

Volume of Frustum Visual Guide

Use the diagram or chart below to understand the concept clearly.

Frustum with Two Circular Bases

Visual guide: Frustum with Two Circular Bases
Larger radius: R Smaller radius: r Perpendicular height: h
A conical frustum has larger radius R, smaller radius r, and perpendicular height h.

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Practice Volume of Frustum Questions

Practise published questions related to this topic.

1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Volume of Frustum Revision Points

Use these formulas and checks while solving frustum volume problems.

  • V = (1/3)πh(R² + Rr + r²).
  • R is the larger radius; r is the smaller radius; h is the perpendicular height.
  • When diameters are given, V = (πh/12)(D² + Dd + d²).
  • Slant height is not used directly in the volume formula.
  • If l is given, h = √(l² - (R - r)²).
  • A frustum becomes a cone when r = 0 and a cylinder when R = r.
  • Volume is written in cubic units.

Related Topic Explainers

Area of Square → Perimeter of Square → Area of Rectangle → Perimeter of Rectangle → Area of Triangle → Perimeter of Triangle → Heron's Formula → Area of Triangle Using Two Sides and Included Angle →

Volume of Frustum FAQs

What is the volume of frustum formula?

The formula is V = (1/3)πh(R² + Rr + r²), where R and r are the larger and smaller radii and h is the perpendicular height.

What is the volume of a frustum with R = 7 cm, r = 4 cm, and h = 5 cm?

V = (1/3)π × 5 × (49 + 28 + 16) = 465π/3 = 155π cm³, approximately 486.92 cm³.

Can slant height be used directly in the frustum volume formula?

No. The formula requires perpendicular height. If slant height l is known, calculate h using h = √(l² - (R - r)²).

What is the frustum volume formula when diameters are given?

For larger diameter D, smaller diameter d, and height h, use V = (πh/12)(D² + Dd + d²).

How does the frustum formula become the cone formula?

Put r = 0 in V = (1/3)πh(R² + Rr + r²). It becomes V = (1/3)πR²h, the volume of a cone.

How does the frustum formula become the cylinder formula?

Put R = r in the formula. Then V = (1/3)πh(3R²) = πR²h, which is the volume of a cylinder.

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