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Area of a sector

The area of a sector is the part of a circle enclosed by two radii and the corresponding arc.

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Student-friendly explanation

A sector is like a pizza slice. Its area depends on the central angle and the radius of the circle. If the angle becomes larger, the sector area also becomes larger because a bigger part of the circle is included.

How to write this in exams

  1. 1

    Start with the exact idea

    The area of a sector is the part of a circle enclosed by two radii and the corresponding arc.

  2. 2

    Then show how to use it

    1. Identify the central angle. 2. Note the radius. 3. Use area of sector = θ/360 × πr². 4. Substitute carefully. 5. Simplify and write square units.

  3. 3

    Add one concrete example

    For a sector of angle 60° in a circle of radius 9 cm, area = 60/360 × π × 9² = 13.5π cm².

  4. 4

    Avoid this incomplete answer

    A common wrong answer is 36π cm² for the 120° sector of radius 6 cm when the student forgets to divide by 360.

Definition

The area of a sector is the part of a circle enclosed by two radii and the corresponding arc.

Example

For a sector of angle 60° in a circle of radius 9 cm, area = 60/360 × π × 9² = 13.5π cm².

Rule to remember

Area of sector = (θ/360) × πr², where θ is the central angle in degrees and r is the radius.

Memory hook

Sector area means fraction of the full circle's area.

Examples and method

Worked example

For θ = 120° and r = 6 cm, area = 120/360 × π × 36 = 12π cm² = 37.68 cm² if π = 3.14.

Method to apply

1. Identify the central angle. 2. Note the radius. 3. Use area of sector = θ/360 × πr². 4. Substitute carefully. 5. Simplify and write square units.

Diagram support

Draw a circle, mark the centre, show the two radii, and label the central angle θ clearly on the sector.

How CBSE asks it

Find the area of a sector in a circle, especially in pizza-slice, fan-blade, or pie-chart style problems.

Avoid common mistakes

Common confusion

Students often use the full circle area formula instead of taking only the fraction given by the angle.

Common wrong answer

A common wrong answer is 36π cm² for the 120° sector of radius 6 cm when the student forgets to divide by 360.

Exam tip

Always check whether the figure is the full circle or only a part of it before using the formula.

Quick check

A sector has central angle 90° in a circle of radius 8 cm. What part of the full circle does it represent?

It represents one-fourth of the full circle, because 90° out of 360° is 1/4. So its area is one-fourth of the area of the complete circle.

Study the area of a sector diagram carefully

Use the labelled diagram to keep area of a sector clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of area of a sector in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on area of a sector.

Revision cue

Revise area of a sector through the labels before writing the answer.

Answer writing and exam use

1-mark answer

The area of a sector is the part of a circle enclosed by two radii and the corresponding arc.

2-mark answer

The area of a sector is the part of a circle enclosed by two radii and the corresponding arc. Area of sector = (θ/360) × πr², where θ is the central angle in degrees and r is the radius. For a sector of angle 60° in a circle of radius 9 cm, area = 60/360 × π × 9² = 13.5π cm².

3-mark answer

A sector is like a pizza slice. Its area depends on the central angle and the radius of the circle. If the angle becomes larger, the sector area also becomes larger because a bigger part of the circle is included. Area of sector = (θ/360) × πr², where θ is the central angle in degrees and r is the radius. For θ = 120° and r = 6 cm, area = 120/360 × π × 36 = 12π cm² = 37.68 cm² if π = 3.14. Find the area of a sector in a circle, especially in pizza-slice, fan-blade, or pie-chart style problems. A common wrong answer is 36π cm² for the 120° sector of radius 6 cm when the student forgets to divide by 360.
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