Volume of Sphere with Radius and Formula Examples

Volume of sphere is the space occupied by a spherical solid. It is calculated using the radius and the value of π. This page explains the volume of sphere formula, diameter-based calculations, unit conversions, scaling properties, and solved volume of sphere problems for quantitative aptitude.

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What Is the Volume of a Sphere?

The volume of a sphere is the total three-dimensional space enclosed by its curved surface. Its formula is V = 4/3πr³, where r is the radius and π is approximately 22/7 or 3.14.

The radius is the distance from the centre of the sphere to any point on its surface. Since radius is cubed, the volume is expressed in cubic units. If the diameter d is given, use r = d/2, so V = πd³/6. For a sphere of radius 3 cm, using π = 3.14, V = 4/3 × 3.14 × 3³ = 113.04 cm³.

Volume of Sphere Video Lessons

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Sphere and Hemisphere: Volume and Surface Area

Learn the formulas and methods to calculate the volume and surface area of spheres and hemispheres, including how to apply them to quantitative aptitude problems.

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Volume of Sphere Formula & Tricks

Important Formulas

Volume using radius
V = 4/3 πr³

Use this formula when the radius r of the sphere is known.

Volume using diameter
V = πd³/6

Use this form when the diameter d is given, because r = d/2.

Radius from volume
r = ∛(3V/4π)

Use the cube root after rearranging V = 4/3πr³.

Diameter from volume
d = ∛(6V/π)

This follows from V = πd³/6.

Quick Tricks

Convert diameter to radius first

When diameter is given, divide it by 2 before applying V = 4/3πr³. Alternatively, use V = πd³/6 directly.

Example: For d = 6 cm, V = π × 6³/6 = 36π cm³ = 113.04 cm³ using π = 3.14.
Use the radius-cube ratio

For spheres of the same material, volumes are proportional to the cubes of their radii. Thus, V₁:V₂ = r₁³:r₂³.

Example: If the radii are in the ratio 2:3, the volumes are in the ratio 2³:3³ = 8:27.
Count smaller spheres after melting

If one large sphere is melted into equal smaller spheres without loss of material, the number of smaller spheres equals the ratio of their volumes.

Example: A sphere of radius 6 cm forms spheres of radius 2 cm. Number = (6/2)³ = 27.

Volume of Sphere Concepts

Using Radius and Diameter

The radius must be used in the formula V = 4/3πr³, while the diameter can be used directly in V = πd³/6.

The diameter is twice the radius: d = 2r. Therefore, replacing r by d/2 gives V = 4/3π(d/2)³ = πd³/6. Always keep the length unit unchanged before cubing it.

Example: For a sphere with diameter 10 cm, V = π × 10³/6 = 500π/3 cm³, approximately 523.33 cm³ using π = 3.14.

Effect of Changing the Radius

The volume changes as the cube of the radius, so V is directly proportional to r³.

If the radius changes by a factor k, the volume changes by a factor k³. Doubling the radius makes the volume 8 times larger, while tripling it makes the volume 27 times larger.

Example: If two spheres have radii 4 cm and 8 cm, their volume ratio is 4³:8³ = 1:8.

Volume Ratio of Two Spheres

For two spheres, the ratio of volumes is V₁:V₂ = r₁³:r₂³.

The common factor 4π/3 cancels when volumes are compared. If diameters are given instead, the same rule applies because diameter and radius have a constant ratio of 2.

Example: For radii 5 cm and 10 cm, V₁:V₂ = 5³:10³ = 125:1000 = 1:8.

Melted Sphere and Equal Smaller Spheres

When a sphere is melted into equal smaller spheres without loss of material, the number of smaller spheres equals (R/r)³.

The volume of the large sphere equals the total volume of the smaller spheres. Thus, (4/3)πR³ = n(4/3)πr³, giving n = (R/r)³.

Example: A sphere of radius 9 cm is melted into spheres of radius 3 cm. The number formed is (9/3)³ = 27.

Volume of Sphere Visual Guide

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

Sphere with Radius Marked

Visual guide: Sphere with Radius Marked
Radius: r = 6 cm Volume: V = 4/3πr³
A sphere is shown with radius r measured from its centre to its surface.

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Practice Volume of Sphere 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 Sphere: Quick Revision

Use these formulas and rules for direct calculations and comparison problems.

  • Volume of a sphere: V = 4/3πr³.
  • Using diameter: V = πd³/6.
  • Radius is half the diameter: r = d/2.
  • Volume is measured in cubic units such as cm³ or m³.
  • If radius changes in the ratio a:b, volume changes in the ratio a³:b³.
  • For a melted sphere divided into equal smaller spheres, number = (large radius/small radius)³.
  • Use π = 22/7 when the calculation allows exact cancellation; otherwise π = 3.14 may be used as specified.

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 Sphere FAQs

What is the volume of sphere formula?

The formula is V = 4/3πr³, where r is the radius. If the diameter d is given, use V = πd³/6.

How do you find the volume of a sphere with diameter 14 cm?

Here r = 7 cm. Therefore, V = 4/3 × 22/7 × 7³ = 4312/3 cm³, or approximately 1437.33 cm³.

What is the volume of a sphere with radius 6 cm using π = 3.14?

V = 4/3 × 3.14 × 6³ = 904.32 cm³.

If the radius of a sphere is doubled, how does its volume change?

The volume becomes 2³ = 8 times the original volume because volume is proportional to the cube of the radius.

What is the volume ratio of spheres whose radii are in the ratio 3:4?

Their volume ratio is 3³:4³ = 27:64.

How many spheres of radius 2 cm can be made by melting a sphere of radius 6 cm?

Number = (6/2)³ = 27 spheres, assuming no material is lost.

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