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Volume of a sphere

The volume of a sphere is the amount of space enclosed inside it.

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Student-friendly explanation

A sphere is completely round, so its volume depends only on the radius. The formula shows that volume grows very fast when radius increases, because the radius is cubed. This makes sphere volume questions especially important in three-dimensional mensuration.

How to write this in exams

  1. 1

    Start with the exact idea

    The volume of a sphere is the amount of space enclosed inside it.

  2. 2

    Then show how to use it

    1. Identify the radius. 2. Cube the radius. 3. Multiply by 4/3 and π. 4. Simplify the result. 5. Write the answer in cubic units.

  3. 3

    Add one concrete example

    A sphere of radius 3 cm has volume 4/3 πr³ = 4/3 × π × 27 = 36π cm³, which is about 113.14 cm³.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to use 4πr². That gives surface area, not the space enclosed inside the sphere.

Definition

The volume of a sphere is the amount of space enclosed inside it.

Example

A sphere of radius 3 cm has volume 4/3 πr³ = 4/3 × π × 27 = 36π cm³, which is about 113.14 cm³.

Rule to remember

Volume of a sphere = 4/3 πr³, where r is the radius of the sphere.

Memory hook

Sphere volume is four-thirds pi times radius cubed.

Examples and method

Worked example

For r = 3 cm, volume = 4/3 πr³ = 4/3 × π × 27 = 36π cm³.

Method to apply

1. Identify the radius. 2. Cube the radius. 3. Multiply by 4/3 and π. 4. Simplify the result. 5. Write the answer in cubic units.

Diagram support

Draw a sphere in a circle-like sectional view and mark the radius from the center to the surface. Remind students that no height or base is used.

How CBSE asks it

Find the capacity or volume of a sphere, ball, globe, or spherical solid using the given radius.

Avoid common mistakes

Common confusion

Students often use 4πr², which is the surface area formula, instead of the volume formula.

Common wrong answer

A common wrong answer is to use 4πr². That gives surface area, not the space enclosed inside the sphere.

Exam tip

For sphere volume, remember the fraction 4/3 and the cube of the radius.

Quick check

A solid ball has radius 3 cm. Which expression gives its volume?

Use 4/3 πr³, because sphere volume depends on the cube of the radius and includes the factor four-thirds.

Answer writing and exam use

1-mark answer

The volume of a sphere is the amount of space enclosed inside it.

2-mark answer

The volume of a sphere is the amount of space enclosed inside it. Volume of a sphere = 4/3 πr³, where r is the radius of the sphere. A sphere of radius 3 cm has volume 4/3 πr³ = 4/3 × π × 27 = 36π cm³, which is about 113.14 cm³.

3-mark answer

A sphere is completely round, so its volume depends only on the radius. The formula shows that volume grows very fast when radius increases, because the radius is cubed. This makes sphere volume questions especially important in three-dimensional mensuration. Volume of a sphere = 4/3 πr³, where r is the radius of the sphere. For r = 3 cm, volume = 4/3 πr³ = 4/3 × π × 27 = 36π cm³. Find the capacity or volume of a sphere, ball, globe, or spherical solid using the given radius. A common wrong answer is to use 4πr². That gives surface area, not the space enclosed inside the sphere.
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