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Area of a segment

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

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

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

Definition

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

Example

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

Rule to remember

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

Memory hook

Segment means sector minus triangle.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

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.

Answer writing and exam use

1-mark answer

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

2-mark answer

The area of a segment is the area enclosed between a chord and the corresponding arc of a circle. Area of segment = area of sector - area of triangle formed by the two radii and the chord. For a minor segment, if sector area is 60 cm² and triangle area is 20 cm², then segment area is 40 cm².

3-mark answer

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. Area of segment = area of sector - area of triangle formed by the two radii and the chord. If radius is 10 cm and central angle is 60°, sector area = 50π/3 cm² and triangle area = 25√3 cm². Segment area = 50π/3 - 25√3 cm². Find the shaded part bounded by a chord and arc, especially when the figure combines sector and triangle geometry. A common wrong answer is to stop at the sector area and ignore the triangle that must be removed.
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