Area from two sides and the included angle
When two sides and the angle between them are known, the area of a triangle is half the product of those sides times the sine of the included angle. This works for any triangle, not just right-angled ones.
Key formula
Area = ½ ab sin C = ½ bc sin A = ½ ac sin B. Heron's formula: Area = √[s(s−a)(s−b)(s−c)], where s = (a + b + c)/2.
Worked example
A triangle has sides a = 6 cm and b = 8 cm with included angle C = 30°. Find its area. Area = ½ ab sin C = ½ × 6 × 8 × sin 30° = ½ × 48 × 0.5 = 24 × 0.5 = 12 cm². The sine of the included angle is essential; the two sides must enclose that angle.
Heron's formula
When only the three sides are known, compute the semi-perimeter s = (a + b + c)/2, then Area = √[s(s−a)(s−b)(s−c)]. For a = 3, b = 4, c = 5: s = 6, Area = √[6(3)(2)(1)] = √36 = 6 cm².
Combining with the other rules
Often you must first find a missing side or angle before the area formula can be applied. If you know two angles and one side, use the sine rule to obtain a second side, then use ½ab sin C. If you know three sides, either use Heron's formula directly or find one angle with the cosine rule and then apply ½ab sin C. Always check that the angle used in ½ab sin C is enclosed by the two chosen sides; using a non-included angle is the most common mistake.
Remember
- The angle in ½ab sin C must be the included angle.
- Area units are square units (cm²).
- Use Heron's formula when all three sides are known.