C
CraftExam
medium difficulty5 min

Length of an arc

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

In one line: Arc is the curved border, not the filled floor.

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Understand it

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.

Example

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

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

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

Method you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer
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.

Common wrong answer

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

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

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.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

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

  3. 3

    Add for 3 marks

    An arc is not the whole circumference.

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