Volume of Cone with Radius and Height Calculations

Volume of Cone is the space occupied by a cone. Its value depends on the base radius and perpendicular height. The standard formula is V = ⅓πr²h. This page explains the formula, units, diameter-based forms, solved volume of cone examples, and methods for solving common mensuration problems accurately.

On this page

What Is the Volume of a Cone?

The volume of a cone is the space enclosed by its circular base and curved surface. It is given by V = ⅓πr²h, where r is the base radius and h is the perpendicular height.

A cone has one circular base and one vertex. To calculate its volume, square the radius, multiply by the perpendicular height and π, then divide by 3. The answer is expressed in cubic units. For r = 3 cm and h = 4 cm, V = ⅓ × π × 3² × 4 = 12π cm³, or approximately 37.7 cm³.

Volume of Cone Video Lessons

Watch short topic-wise lessons for quick revision.

8 Lessons
Lesson 1 of 8 Quick Revision

Cone Volume and Surface Area

Learn the formulas for a cone’s volume, slant height, curved surface area, and total surface area, with guidance on applying them to mensuration problems.

Continue with more lessons and practice in PrepShots.Watch More in App - Start ₹1 Trial →
More Volume of Cone Lessons Scroll to explore →

Volume of Cone Formula & Tricks

Important Formulas

Standard volume formula
V = ⅓πr²h

Here, r is the radius of the circular base and h is the perpendicular height of the cone.

Using diameter
V = πd²h ÷ 12

Since r = d/2, substitute r² = d²/4 in V = ⅓πr²h, where d is the diameter.

Finding height
h = 3V ÷ (πr²)

Rearrange the volume formula when the volume and radius are known.

Finding radius
r = √(3V ÷ πh)

Rearrange the volume formula when the volume and perpendicular height are known.

Quick Tricks

Use the diameter carefully

The formula uses radius, not diameter. If the diameter is given, first use r = d/2 or directly apply V = πd²h ÷ 12.

Example: For d = 6 cm and h = 5 cm, V = π × 6² × 5 ÷ 12 = 15π cm³.
Compare volumes by ratios

For cones with the same height, volume is proportional to r². For cones with the same radius, volume is proportional to h.

Example: If the radius changes from r to 2r while height remains fixed, the volume becomes 4 times the original volume.
Keep units consistent

Convert all dimensions to the same unit before substitution. The final unit is cubic: cm³, m³ or another length unit cubed.

Example: A radius of 50 cm and height of 2 m should be converted to r = 0.5 m before calculating volume in m³.

Volume of Cone Concepts

Radius, Diameter and Perpendicular Height

The volume formula requires the base radius and the perpendicular height, not the slant height.

The radius is the distance from the centre of the circular base to its edge. The diameter is twice the radius. The perpendicular height is the shortest distance from the vertex to the plane of the base. If slant height l and radius r are given, the height can be found using h = √(l² − r²).

Example: If l = 13 cm and r = 5 cm, then h = √(13² − 5²) = √144 = 12 cm.

Applying the Volume Formula

Substitute the radius and perpendicular height in V = ⅓πr²h, then simplify the result with the required value of π.

The radius must be squared before multiplication. Use π = 22/7 when the dimensions make cancellation convenient, or π = 3.14 when specified. For r = 7 cm and h = 12 cm, V = ⅓ × 22/7 × 49 × 12 = 616 cm³.

Example: For r = 7 cm and h = 12 cm, the volume is 616 cm³ using π = 22/7.

Volume Ratios of Cones

The ratio of cone volumes is V₁:V₂ = r₁²h₁:r₂²h₂ when both cones use the same value of π.

The common factor ⅓π cancels in a comparison. Thus, if the radii are equal, volumes are in the ratio of heights. If heights are equal, volumes are in the ratio of the squares of radii.

Example: For two cones with radii 2 cm and 3 cm and equal heights, V₁:V₂ = 2²:3² = 4:9.

Capacity and Unit Conversion

The numerical volume represents capacity only when the solid is a container, and the units must be converted correctly.

Useful conversions include 1 m³ = 1,000 litres, 1 litre = 1,000 cm³, and 1 cm³ = 1 millilitre. Therefore, a volume in cubic metres is multiplied by 1,000 to express it in litres.

Example: A conical container with volume 0.5 m³ has capacity 0.5 × 1,000 = 500 litres.

Volume of Cone Visual Guide

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

Cone with radius and perpendicular height

Visual guide: Cone with radius and perpendicular height
Radius: r Perpendicular height: h
The volume depends on the circular base radius r and perpendicular height h.

Learn more concepts on PrepShots

Start ₹1 Trial →

Practice Volume of Cone 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?→

Want more practice questions?

Practice in App →

Volume of Cone Quick Quiz

Attempt 5 questions and check your score instantly.

Quick Revision Notes

Volume of Cone: Revision Points

Use these formulas and rules for quick revision.

  • Volume of a cone is V = ⅓πr²h.
  • Use perpendicular height, not slant height, in the volume formula.
  • If diameter d is given, use V = πd²h ÷ 12.
  • The answer has cubic units.
  • For equal heights, cone volumes are proportional to the squares of their radii.
  • Use h = √(l² − r²) when slant height and radius are given.
  • 1 m³ = 1,000 litres and 1 litre = 1,000 cm³.

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 Cone FAQs

What is the volume of cone formula?

The formula is V = ⅓πr²h, where r is the base radius and h is the perpendicular height.

How do you find the volume of a cone when its diameter is given?

Use V = πd²h ÷ 12, where d is the diameter and h is the perpendicular height. This follows from r = d/2.

What is the volume of a cone with radius 3 cm and height 4 cm?

V = ⅓ × π × 3² × 4 = 12π cm³, approximately 37.7 cm³.

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

No. The formula uses perpendicular height. If slant height l and radius r are known, calculate h = √(l² − r²) first.

What happens to the volume if the radius is doubled and height remains unchanged?

The volume becomes four times the original because volume is proportional to r² when height is fixed.

How many litres are equal to 0.25 cubic metre of cone volume?

Since 1 m³ = 1,000 litres, 0.25 m³ = 250 litres.

Continue learning Volume of Cone on PrepShots

Continue on PrepShots