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Length of an arc

The length of an arc is the part of the circle's boundary between two points on the circle.

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Student-friendly explanation

An arc is not the whole circumference. It is only a portion of the boundary, so its length is a fraction of 2πr depending on the central angle. This idea is very common in sector and shaded-region questions.

How to write this in exams

  1. 1

    Start with the exact idea

    The length of an arc is the part of the circle's boundary between two points on the circle.

  2. 2

    Then show how to use it

    1. Identify the angle subtended at the centre. 2. Identify the radius. 3. Use arc length = θ/360 × 2πr. 4. Simplify carefully. 5. State the answer as a length.

  3. 3

    Add one concrete example

    If a circle has radius 10 cm and a central angle of 90°, the arc length is 90/360 × 2π × 10 = 5π cm.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is 10π cm² when the student writes square units for a length quantity.

Definition

The length of an arc is the part of the circle's boundary between two points on the circle.

Example

If a circle has radius 10 cm and a central angle of 90°, the arc length is 90/360 × 2π × 10 = 5π cm.

Rule to remember

Arc length = (θ/360) × 2πr, where θ is the central angle in degrees and r is the radius.

Memory hook

Arc is the curved border, not the filled floor.

Examples and method

Worked example

For θ = 150° and r = 12 cm, arc length = 150/360 × 2π × 12 = 10π cm = 31.4 cm.

Method to apply

1. Identify the angle subtended at the centre. 2. Identify the radius. 3. Use arc length = θ/360 × 2πr. 4. Simplify carefully. 5. State the answer as a length.

Diagram support

Draw a sector and highlight only the curved outer boundary to show what arc length means.

How CBSE asks it

Find the curved part of a sector, or use arc length to help in perimeter of sector and shaded region questions.

Avoid common mistakes

Common confusion

Students often use sector area formula instead of arc length formula because both use the same angle fraction.

Common wrong answer

A common wrong answer is 10π cm² when the student writes square units for a length quantity.

Exam tip

If the question says boundary part, curved edge, or only the arc, use arc length, not area.

Quick check

A 60° sector is drawn in a circle. What does the arc length refer to?

The arc length is the curved part of the boundary of that sector. It measures how long the circular edge is between the two radii, not the space inside the sector.

Answer writing and exam use

1-mark answer

The length of an arc is the part of the circle's boundary between two points on the circle.

2-mark answer

The length of an arc is the part of the circle's boundary between two points on the circle. Arc length = (θ/360) × 2πr, where θ is the central angle in degrees and r is the radius. If a circle has radius 10 cm and a central angle of 90°, the arc length is 90/360 × 2π × 10 = 5π cm.

3-mark answer

An arc is not the whole circumference. It is only a portion of the boundary, so its length is a fraction of 2πr depending on the central angle. This idea is very common in sector and shaded-region questions. Arc length = (θ/360) × 2πr, where θ is the central angle in degrees and r is the radius. For θ = 150° and r = 12 cm, arc length = 150/360 × 2π × 12 = 10π cm = 31.4 cm. Find the curved part of a sector, or use arc length to help in perimeter of sector and shaded region questions. A common wrong answer is 10π cm² when the student writes square units for a length quantity.
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