Triangle area, sectors & segments
Three area formulas cover almost every case. With two sides and the included angle, Area = ½ab sin C. With three sides, Heron’s formula uses the semi-perimeter s: √(s(s−a)(s−b)(s−c)). For circles, a sector is ½r²θ and a segment is the sector minus the triangle, ½r²(θ − sin θ) — with θ in radians throughout.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Match the formula to what you knowTwo sides plus INCLUDED angle → ½ab sin C. Three sides → Heron. Base and height → ½bh.
- Compute the semi-perimeter first for Herons = (a+b+c)/2, then apply the product under the root.
- Use radians for sectors and segmentsA = ½r²θ needs radians; converting first avoids an answer wrong by a factor of 57.
- Segment = sector − triangle½r²θ − ½r² sin θ, which factors to ½r²(θ − sin θ).
Worked example
Find the area of a segment in a circle of radius 10 cut off by a chord subtending 90° at the centre.
- Convert: 90° = π/2 radians.
- Sector area: ½r²θ = ½ × 100 × π/2 = 25π ≈ 78.54.
- Triangle area: ½r² sin θ = ½ × 100 × sin(π/2) = 50.
- Segment = 25π − 50.
Answer. Segment area = 25π − 50 ≈ 28.54 square units.
Where marks get dropped
These are the specific errors that cost credit on triangle area, sectors & segments questions — QED's rubric penalises each of them separately.
- Using ½ab sin C with an angle that is not between the two sides. The angle must be the included one.
- Using degrees in the sector formula, which inflates the answer enormously.
- Forgetting to subtract the triangle when a segment is asked for rather than a sector.
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Triangle area, sectors & segments — frequently asked questions
Why does ½ab sin C work?
Because the height from the vertex is b sin C, so ½ × base × height = ½ab sin C. It generalises ½bh to non-right triangles.
When should I use Heron’s formula?
When you know all three sides and no angles. Otherwise ½ab sin C is quicker and less error-prone.
What is the semi-perimeter?
Half the perimeter, s = (a+b+c)/2. It appears in Heron’s formula and in the inradius relation Area = rs.
The rest of Trigonometry
Triangles, the unit circle, identities and periodic functions. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Right-triangle ratios: sin, cos & tan
- 2The unit circle & exact values
- 3Radians, degrees & arc length
- 4Graphs of sin, cos & tan
- 5Pythagorean & reciprocal identities
- 6Angle-sum, difference & double-angle formulas
- 7Solving trigonometric equations
- 8The sine rule & the cosine rule
- 9Triangle area, sectors & segments
- 10Inverse trigonometric functions
- 11Proving trigonometric identities
- 12Polar coordinates & converting to Cartesian
- 13Modelling periodic phenomena with sinusoids
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