A chord subtends an angle of 60^ at the centre of a circle of radius 10 cm. Find the area of the minor segment.

Sector area =(60)/(360)pi(100) =52.36. Triangle area =12 r^2=12(100)60^ =43.30. Segment =52.36-43.30 is approximately 9.06 cm². Set the angle unit first (degrees unless the question uses radians), then use sin/cos/tan and their inverses.

A washer is the region between two sectors of the same angle 90^, with inner radius 3 cm and outer radius 5 cm. Find its area.

Outer sector =(90)/(360)pi(25)=19.63. Inner sector =(90)/(360)pi(9)=7.07. Area =19.63-7.07 is approximately 12.6 cm². A GDC is permitted on this paper, so you may evaluate or verify this result directly on the calculator.

Segment Area (AI SL)

Q: What is the formula for the area of a segment?

segment = sector - (1/2)r^2 sin(theta), where theta is the central angle in degrees and r is the radius. It's the sector area with the triangle formed by the two radii and the chord subtracted off.

A segment is what's left of a sector once you cut off the triangle formed by the two radii and the chord. Where a sector is a full "pie slice", a segment is the curved sliver beyond a straight chord - and composite shapes built from circles usually reduce to adding or subtracting a segment somewhere. This page covers the method with worked examples and the mistakes that lose the most marks. It's part of the broader Arcs & Sectors topic.

11 questions on this sub-topic.

Practise segment area → Try exam-style questions

Sector minus triangle

Covered under IB syllabus reference SL3.4: length of an arc and area of a sector, extended here to the segment that sits beyond the chord. Radians aren't required at SL - the central angle is always used as a fraction of \(360^\circ\).

Sector area

\[A=\dfrac{\theta}{360}\times\pi r^2\]

The fraction of the circle's total area swept out by the central angle.

✓ In the formula booklet

Segment area

\[\text{segment}=\dfrac{\theta}{360}\pi r^2-\tfrac12 r^2\sin\theta\]

Sector area minus the triangle formed by the two radii and the chord.

Not in the formula booklet - combine two results

Need the full syllabus wording and formula-booklet reference table? See Arcs & Sectors.

Worked examples

Hard

[5 marks]

A chord subtends an angle of \(60^\circ\) at the centre of a circle of radius 10 cm.

Find the area of the minor segment.

Worked solution

Sector area \(=\dfrac{60}{360}\pi(100)\) M1
\(=52.36.\) A1
Triangle area \(=\tfrac12 r^2\sin\theta=\tfrac12(100)\sin60^\circ\) M1
\(=43.30.\) A1
Segment \(=52.36-43.30\approx9.06\) cm². A1

Set the angle unit first (degrees unless the question uses radians), then use sin/cos/tan and their inverses.

M1 Sector A1 Correct answer of 52.36 M1 Triangle A1 Correct answer of 43.30 A1 Correct answer of 9.06

Hard

[4 marks]

A washer is the region between two sectors of the same angle \(90^\circ\), with inner radius 3 cm and outer radius 5 cm.

Find its area.

Worked solution

Outer sector \(=\dfrac{90}{360}\pi(25)=19.63.\) M1
Inner sector \(=\dfrac{90}{360}\pi(9)=7.07.\) A1
Area \(=19.63-7.07\) M1
\(\approx12.6\) cm². A1

A GDC is permitted on this paper, so you may evaluate or verify this result directly on the calculator.

M1 Outer A1 Inner M1 Subtract A1 Correct answer of 12.6

Common mistakes

Forgetting to subtract the triangle. A sector is the full pie-slice; a segment is what's left once the triangle formed by the two radii and the chord is cut away - stopping at the sector area is the single most common error here.
Using \(r^2\) instead of the two different radii in a composite "washer" shape. When a shape is the region between two concentric sectors, each sector needs its own radius squared before you subtract - don't reuse one radius for both.
Leaving the calculator in the wrong angle mode. \(\sin\theta\) in the triangle-area term needs degree mode when \(\theta\) is in degrees; radian mode silently gives a completely wrong triangle area with no error message.

Full topicArcs & Sectors RelatedArc Length and Sector Area FormulasAI SL formula booklet explained PrintAI SL formula sheet

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11 segment-area questions, marked instantly like the real exam.

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Quick answers

What is the formula for the area of a segment?

\(\text{segment}=\dfrac{\theta}{360}\pi r^2-\tfrac12 r^2\sin\theta\), where \(\theta\) is the central angle in degrees and \(r\) is the radius. It's the sector area with the triangle formed by the two radii and the chord subtracted off.

Is the segment area formula in the formula booklet?

No. The formula booklet gives sector area and the triangle-area formula \(\tfrac12 ab\sin C\) separately - you combine the two yourself to get the segment area. See the GDC pointers on the Arcs & Sectors page for calculator setup.

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