medium difficulty6 min

Area of a segment

The area of a segment is the area enclosed between a chord and the corresponding arc of a circle.

In one line: Segment means sector minus triangle.

Understand it

A segment is a smaller part of a circle cut off by a chord. To find its area, students usually find the sector area first and then subtract the area of the triangle formed by the two radii and the chord. This is a common exam method.

Example

For a minor segment, if sector area is 60 cm² and triangle area is 20 cm², then segment area is 40 cm².

Formula and rule

Area of segment = area of sector - area of triangle formed by the two radii and the chord.

See it worked out

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

Worked example

1If radius is 10 cm and central angle is 60°, sector area = 50π/3 cm² and triangle area = 25√3 cm².
2Segment area = 50π/3 - 25√3 cm².

Method you can reuse

1Identify whether the segment is minor or major.
2Find the sector area using angle and radius.
3Find the triangle area if the radii and chord form a triangle.
4Subtract carefully.
5Write the answer in square units.

Sketch it in your answer

Draw the chord inside the circle, mark the radii to the chord endpoints, and show the small region between chord and arc.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

A chord and an arc form a small region in a circle. What do you do first to find its area?

Think first, then show answer

You usually find the sector area first and then subtract the area of the triangle formed by the radii and the chord. That gives the segment area.

Common wrong answer

A common wrong answer is to stop at the sector area and ignore the triangle that must be removed.

Don't mix this up

Students often forget to subtract the triangle area and write the sector area as the final answer.

Exam tip

Whenever a chord is inside a circle, check whether the question is asking for the smaller segment or the larger part.

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 shaded part bounded by a chord and arc, especially when the figure combines sector and triangle geometry.

1
Enough for 1 mark
The area of a segment is the area enclosed between a chord and the corresponding arc of a circle.
2
Add for 2 marks
Example: For a minor segment, if sector area is 60 cm² and triangle area is 20 cm², then segment area is 40 cm².
3
Add for 3 marks
A segment is a smaller part of a circle cut off by a chord.

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