5.3PureStretch

Areas of Sectors and Segments

In radians the area of a sector is a clean fraction of the whole circle. Subtracting the triangle from the sector gives the area of a segment — a common exam question.

30 min Video by Zeeshan Zamurred Radians

Edexcel A Level Maths: 5.3 Area of Sectors and SegmentsWatch the full walkthrough before the notes below.

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What you'll be able to do

Use A = ½r²θ for sector area
Find the area of a segment
Combine with arc length results
Solve problems with unknown r or θ

Area of a sector

For radius $r$ and angle $θ$ in radians, the sector area is $A = \frac{1}{2} r^{2} θ$ . (It is the fraction $\frac{θ}{2 π}$ of the full circle area $π r^{2}$ .)

A = \frac{1}{2} r^{2} θ (θ in radians)

Sector area.

Area of a segment

A segment is the region between a chord and the arc. Its area is the sector minus the triangle formed by the two radii: $A = \frac{1}{2} r^{2} θ - \frac{1}{2} r^{2} sin θ = \frac{1}{2} r^{2} (θ - sin θ)$ .

A_{segment} = \frac{1}{2} r^{2} (θ - sin θ)

Sector minus triangle.

\frac{1}{2} (6)^{2} (\frac{π}{3} - sin \frac{π}{3})

= 18 (1.047 - 0.866) \approx 18 (0.181)

Answer

\approx 3.26

cm²

Tip — Triangle area between two radii is ½r²sin θ (using two sides and the included angle).

Formula recap

A = \frac{1}{2} r^{2} θ

Sector area.

A = \frac{1}{2} r^{2} (θ - sin θ)

Segment area.

Common mistakes to avoid

Using degrees in A = ½r²θ.

θ must be in radians.

Forgetting to subtract the triangle for a segment.

Segment = sector − ½r²sin θ.

Key takeaways

Sector area: A = ½r²θ (θ in radians).
Segment area: ½r²(θ − sin θ).
Triangle between two radii: ½r²sin θ.

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