medium difficulty6 min

Difference between arc length and sector area

Arc length is the curved boundary part of a sector, while sector area is the space enclosed by the sector.

In one line: Arc is line, sector is space.

Practice This Concept

Understand it

These two quantities use the same central angle and radius, but they measure different things.
Arc length is a length, so its unit is linear.
Sector area is a surface measure, so its unit is square.
Students must read the question carefully to avoid mixing them.

Example

For a 90° sector of radius 8 cm, arc length is 4π cm, but sector area is 16π cm².

Formula and rule

Arc length = (θ/360) × 2πr.

Sector area = (θ/360) × πr².

One is length, the other is area.

See it worked out

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

Worked example

1For θ = 60° and r = 9 cm, arc length = 3π cm, while sector area = 13.5π cm².
2The two answers are different because they measure different quantities.

Method you can reuse

1Read whether the question asks for boundary or inside region.
2Choose arc length or sector area accordingly.
3Substitute the angle and radius.
4Keep units correct.
5Compare the result if needed.

Visual tip

In the sketch, circle the curved boundary for arc length and shade the interior for sector area.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

A student is asked to find the curved edge of a sector. Should they use arc length or sector area?

Think first, then show answer

They should use arc length, because the curved edge is a boundary measure. Sector area would be used only if the question asked for the space inside the sector.

Common wrong answer

A common wrong answer is to use πr² for arc length or to write cm² for the arc answer.

Don't mix this up

Students often use the sector area formula when the question asks for the curved boundary, or they write square units for a length.

Exam tip

Ask yourself: is the question asking for border or inside space? Border means arc length; inside means sector 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

Questions may ask for only the arc, only the sector, or ask you to compare both values from the same figure.

1
Enough for 1 mark
Arc length is the curved boundary part of a sector, while sector area is the space enclosed by the sector.
2
Add for 2 marks
Example: For a 90° sector of radius 8 cm, arc length is 4π cm, but sector area is 16π cm².
3
Add for 3 marks
These two quantities use the same central angle and radius, but they measure different things.

MCQ Quiz

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