When you know two sides of a triangle and the angle between them, the area formula is Area = ½ab sin C, where a and b are the two known sides and C is the included angle. This works for any triangle — right-angled, acute, or obtuse — without needing the perpendicular height.
Why can't you always use Area = ½ × base × height?
The standard area formula requires the perpendicular height, which is often not given and is awkward to calculate for non-right-angled triangles. The sine formula bypasses this by expressing the height in terms of the angle.
In triangle ABC, if you draw the perpendicular height h from vertex B to side AC, then h = a sin C (from right-triangle trigonometry). Substituting into ½ × base × height:
Area = ½ × b × a sin C = ½ab sin C
This derivation means the sine formula is simply the standard formula in disguise, using trigonometry to eliminate the unknown height.
What do the letters in Area = ½ab sin C represent?
- a and b are any two sides of the triangle.
- C is the angle between those two sides (the included angle).
- The angle and its two sides form the letter A shape at vertex C.
If you label the sides differently — for example, p, q and angle R — the formula becomes Area = ½pq sin R. The key is always: two sides and the angle sandwiched between them.
How do you calculate the area when the angle is acute?
Worked example 1
Triangle PQR has PQ = 8 cm, PR = 11 cm, and angle P = 36°. Find the area.
Area = ½ × PQ × PR × sin P
Area = ½ × 8 × 11 × sin 36°
Area = ½ × 8 × 11 × 0.5878…
Area = 44 × 0.5878
Area ≈ 25.9 cm² (3 significant figures)
Always show the substitution clearly — exam mark schemes award method marks for the formula written with values.
Worked example 2
A triangle has sides 7.5 cm and 12 cm with an included angle of 54°. Find the area.
Area = ½ × 7.5 × 12 × sin 54°
= ½ × 90 × 0.8090…
= 45 × 0.8090
Area ≈ 36.4 cm²
How do you calculate the area when the angle is obtuse?
The formula works unchanged for obtuse angles. Your calculator gives sin of angles between 90° and 180° directly. Note that sin(θ) = sin(180° − θ), so sin 120° = sin 60° ≈ 0.866.
Worked example 3
Triangle ABC has AB = 9 cm, BC = 6 cm, and angle B = 130°. Find the area.
Area = ½ × 9 × 6 × sin 130°
sin 130° = sin 50° ≈ 0.7660
Area = ½ × 54 × 0.7660
Area = 27 × 0.7660
Area ≈ 20.7 cm²
The presence of an obtuse angle does not change the method — simply enter the obtuse angle into your calculator as normal.
How do you find a side or angle when the area is given?
Rearrange the formula:
- To find a missing side: a = (2 × Area) ÷ (b × sin C)
- To find a missing angle: sin C = (2 × Area) ÷ (ab), then C = sin⁻¹ of the result
Worked example 4: find the angle
A triangle has area 40 cm², with two sides of length 10 cm and 12 cm. Find the included angle.
sin C = (2 × 40) ÷ (10 × 12) = 80 ÷ 120 = 2/3
C = sin⁻¹(2/3) ≈ 41.8° (or 138.2° — two solutions since sin is positive in both first and second quadrants; context determines which is valid)
Summary of the formula and when to use it
| Known information | Formula to use |
|---|---|
| Base and perpendicular height | Area = ½ × base × height |
| Two sides and included angle | Area = ½ab sin C |
| Three sides given (no angle) | Heron's formula (not GCSE) |
| On the GCSE formula sheet? | Yes — it is given to you |
Frequently asked questions
Is Area = ½ab sin C given on the GCSE formula sheet?
Yes. It appears on both AQA and Edexcel formula sheets. You still need to know which values to substitute and how to rearrange it — but you do not need to memorise it.
Does it matter which two sides I call a and b?
No, provided C is the angle between them. ½ × 8 × 11 × sin 36° gives the same answer as ½ × 11 × 8 × sin 36°. Multiplication is commutative. What you must not do is pick an angle that is NOT between the two chosen sides.
Can I use this formula for a right-angled triangle?
Yes — it simply reduces to the familiar formula. If C = 90°, then sin 90° = 1, giving Area = ½ × a × b × 1 = ½ab. For a right-angled triangle, the two sides adjacent to the right angle are the base and perpendicular height, so this matches ½ × base × height exactly.
How is this formula related to the sine rule and cosine rule?
All three — the sine rule, cosine rule, and this area formula — apply to any triangle. They form a toolkit for solving triangles when you do not have a right angle. The area formula uses the same sine function as the sine rule, and the two sides used are the same pair as in the cosine rule when the included angle is known.
To practise GCSE trigonometry and geometry with personalised hints — visit aitutors.me.