Volume of a Sphere: Formula, Derivation, Worked Examples and Common Mistakes

Quick answer: The volume of a sphere is V = (4/3)πr³, where r is the radius. For a sphere of radius 7 cm, V = (4/3) × (22/7) × 7 × 7 × 7 = 1,437.33 cm³ (about 1437⅓ cm³). A hemisphere is half of that: V = (2/3)πr³.

The answer is always in cubic units (cm³, m³). If you're given the diameter, halve it first. Forgetting that step is the single most common way students lose marks on this formula.

The formulas at a glance

Don't confuse these with the surface area formulas: a sphere's surface area is 4πr², and a solid hemisphere's total surface area is 3πr² (curved 2πr² plus the flat circle πr²). Area has r², volume has r³.

Also read: Surface Area of a Cylinder: Formula, Worked Examples and Practice Questions

Which value of π should you use?

• Use 22/7 when the radius is a multiple of 7 (7, 14, 21, 3.5, 10.5). NCERT questions are usually built this way so the 7s cancel.

• Use 3.14 when the question says so, or when the radius isn't a multiple of 7.

• If a question says "leave your answer in terms of π", don't substitute at all. For r = 3 cm, V = (4/3)π × 27 = 36π cm³.

Always use the value the question asks for. Using 3.14 where 22/7 was given changes the decimals and can cost you the mark in an objective question.

Where the formula comes from (without calculus)

The full proof needs integration, which comes in Class 12. At school level, there are two neat ways to see why (4/3)πr³ is right.

Archimedes' sphere and cylinder

Put a sphere of radius r snugly inside a cylinder of the same radius and height 2r. The Greek mathematician Archimedes showed, in his work On the Sphere and Cylinder, that the sphere takes up exactly two-thirds of the cylinder. He was so proud of this result that he asked for a sphere inside a cylinder to be carved on his tomb.

Check it: cylinder volume = πr² × 2r = 2πr³. Two-thirds of that is (4/3)πr³.

The cone, hemisphere and cylinder activity

Take a cylinder, a cone and a hemisphere, all with radius r, with the cylinder and cone of height r.

• Cylinder: πr² × r = πr³

• Cone: (1/3)πr² × r = (1/3)πr³

• So the hemisphere must hold the rest: πr³ − (1/3)πr³ = (2/3)πr³

You can test this with water: a hemispherical bowl filled with water, poured into a cone of the same radius and height r, fills it twice. Doubling the hemisphere gives the sphere, (4/3)πr³.

Worked examples

Example 1: Volume from the radius

Find the volume of a sphere of radius 3.5 cm.

V = (4/3) × (22/7) × 3.5³ = (4/3) × (22/7) × 42.875 = (4/3) × 134.75 ≈ 179.67 cm³

Example 2: Volume from the diameter

A ball has a diameter of 42 cm. Find its volume.

Radius = 42 ÷ 2 = 21 cm. V = (4/3) × (22/7) × 21 × 21 × 21 = 4 × 22 × 21 × 21 = 38,808 cm³

Example 3: Radius from the volume (reverse question)

A sphere has a volume of 38,808 cm³. Find its radius.

(4/3) × (22/7) × r³ = 38,808 r³ = 38,808 × 3 × 7 ÷ (4 × 22) = 9,261 r = ∛9,261 = 21 cm

Learn the cubes up to 21³ (at least 7³ = 343, 14³ = 2,744, 21³ = 9,261); they turn up again and again in these questions.

Example 4: Hemispherical bowl in litres

How much water can a hemispherical bowl of inner radius 10.5 cm hold?

V = (2/3) × (22/7) × 10.5³ = (2/3) × (22/7) × 1,157.625 = 2,425.5 cm³ Since 1,000 cm³ = 1 litre, the bowl holds about 2.43 litres.

Example 5: Hollow sphere

A hollow metal ball has an outer radius of 7 cm and an inner radius of 3.5 cm. Find the volume of metal.

V = (4/3) × (22/7) × (7³ − 3.5³) = (4/3) × (22/7) × (343 − 42.875) = 1,257.67 cm³ (approx.)

Example 6: What happens when the radius doubles?

Volume depends on r³, so doubling the radius multiplies the volume by 2³ = 8. Tripling it multiplies the volume by 27. The same idea gives ratios: two spheres with radii in the ratio 1 : 2 have volumes in the ratio 1 : 8.

Real-life examples

• Cricket ball. The MCC Laws of Cricket (Law 4) set the circumference of a men's new ball at 22.4–22.9 cm. Taking 22.9 cm: r = 22.9 ÷ 2π ≈ 3.64 cm, so V ≈ (4/3)π(3.64)³ ≈ 203 cm³.

• Football. The IFAB Laws of the Game (Law 2) set a match ball's circumference at 68–70 cm. At 70 cm, r ≈ 11.14 cm and V ≈ 5,790 cm³, about 5.8 litres measured from the outside.

• The Earth. NASA gives Earth's mean radius as 6,371 km. Treating it as a sphere, V ≈ (4/3)π(6,371)³ ≈ 1.08 × 10¹² km³, which matches NASA's figure of 108.321 × 10¹⁰ km³.

Also read: Best Mock Test Platform in India: Five Options Compared

Common mistakes that cost marks

1. Using the diameter as the radius. If the question says "diameter 14 cm", r is 7 cm. Using 14 makes your answer eight times too big.

2. Squaring instead of cubing. r² is for area. Volume always has r³.

3. Mixing up the hemisphere formulas. Volume is (2/3)πr³. Total surface area is 3πr². They look similar on a crowded answer sheet.

4. Wrong units. Volume is in cm³ or m³, not cm². For capacity, convert: 1,000 cm³ = 1 litre and 1 m³ = 1,000 litres.

5. Rounding too early. Keep fractions until the last step; round once at the end.

Where it is in your NCERT book

• Class 9: the new NCERT book Ganita Manjari (Part 2), introduced from the 2026–27 session, covers spheres and hemispheres in Chapter 14, "Math of Space: Surface Area and Volume". Students using the earlier rationalised textbook will find it in Chapter 11, "Surface Areas and Volumes".

• Class 10: in the current textbook (used in 2026–27), Chapter 12, "Surface Areas and Volumes", uses the sphere and hemisphere in combination-of-solids problems, such as a toy made of a cone on a hemisphere, or a cylinder with hemispherical ends. The sections on converting one solid into another and on the frustum of a cone were removed in the 2023 rationalisation. NCERT plans a new Class 10 book from the 2027–28 session, so chapter numbers may change.

For more practice with round solids, see our guide to the surface area of a cylinder. If you're preparing for boards, our tips on paper presentation show how to set out each step so the examiner can award method marks.

Frequently asked questions

What is the formula for the volume of a sphere?

V = (4/3)πr³, where r is the radius. If you know the diameter d, you can also use V = (1/6)πd³.

What is the volume of a hemisphere?

A solid hemisphere's volume is (2/3)πr³, exactly half the volume of a sphere with the same radius.

How do you find the radius if the volume is given?

Rearrange the formula: r³ = 3V ÷ 4π, then take the cube root. With π = 22/7, this often gives a whole number such as 7, 14 or 21.

Why is there a 4/3 in the formula?

It comes from Archimedes' result that a sphere fills two-thirds of the cylinder around it. Two-thirds of the cylinder's volume, 2πr³, is (4/3)πr³.

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