Area of a Triangle Using ½ab sin C
Grade 6The formula
The standard area formula ½ × base × height only works easily when you have a perpendicular height. The sine area formula bypasses that — it uses two sides and the included angle directly.
Area = ½ × a × b × sin C
Here a and b are two sides of the triangle, and C is the angle between them.
How it connects to the standard formula
The perpendicular height h = a sin C (trig in a right-angled triangle). Substitute into ½ × b × h and you get ½ab sin C. So the formula is really the same idea — trig gives you the height you couldn't measure directly.
If C = 90°, sin C = 1, and the formula becomes ½ab — exactly the standard formula for a right-angled triangle. Good check that the formula is correct.
Worked example
EXAMPLE
A triangle has sides a = 8 cm, b = 11 cm, and included angle C = 40°. Find the area.
Area = ½ × 8 × 11 × sin 40° = ½ × 88 × 0.6428 = 44 × 0.6428 ≈ 28.3 cm² (to 3 s.f.).
