Triangle area: ½ab sin C
When you know two sides and the included angle, the area of any triangle (right-angled or not) is:
Area = ½ × a × b × sin C
where a and b are the two sides and C is the angle BETWEEN them (the "included" angle).
Why this works
Drop a perpendicular from the vertex opposite C to side a (or b). Its length is b sin C (using SOH on the small right-angled triangle). Standard triangle area = ½ × base × height = ½ × a × (b sin C).
✦Worked example— Worked examples
Find area. Triangle with sides 6 cm and 9 cm, included angle 50°.
- Area = ½ × 6 × 9 × sin 50° = 27 × 0.766 ≈ 20.69 cm².
Find included angle. Triangle has sides 7 and 10, area 25 cm². Find C.
- 25 = ½ × 7 × 10 × sin C → sin C = 50/70 = 5/7.
- C = sin⁻¹(5/7) ≈ 45.6°.
Find side. Triangle has area 36 cm², one side 8 cm, included angle 60°. Find the other side.
- 36 = ½ × 8 × b × sin 60°.
- 36 = 4b × √3/2 = 2b√3.
- b = 18/√3 = 6√3 ≈ 10.39 cm.
Multi-side application
For larger figures (parallelograms, quadrilaterals divided by a diagonal), apply ½ab sin C to each triangle.
Worked example: parallelogram has sides 5 and 7, included angle 60°. Area = 5 × 7 × sin 60° = 35 × √3/2 ≈ 30.31 cm² (note: parallelogram area = ab sin C, not ½).
Combining with sine/cosine rules
Often used in compound exam questions:
- Use cosine rule to find an angle from three sides.
- Use ½ab sin C with that angle to find area.
⚠Common mistakes
- Using a non-included angle — the angle must be between the two sides quoted.
- Forgetting the ½ — area is HALF the product.
- Calculator mode — degrees for GCSE.
- Mixing sin with cos — use SIN of the included angle.
- Doubling instead of halving when going from triangle to parallelogram.
➜Try this— Quick check
A triangle has sides 4 cm and 5 cm and included angle 30°. Area?
- ½ × 4 × 5 × sin 30° = 10 × 0.5 = 5 cm².
AI-generated · claude-opus-4-7 · v3-deep-geometry