Area of a triangle

The formula

• Area = ½ ab sin C, where a and b are two sides and C is the angle between them.
• Example: a = 5, b = 7, C = 60° → Area = ½(5)(7)(sin 60°) = 17.5 × 0.866 ≈ 15.16.
• Example: b = 10, c = 12, A = 30° → Area = ½(10)(12)(0.5) = 30.
• Heron's formula for three sides: Area = √(s(s − a)(s − b)(s − c)), where s = (a + b + c)/2.
• Example: sides 5, 6, 7 → s = 9 → Area = √(9·4·3·2) = √216 ≈ 14.70.

Choosing the right height

• The classic ½ × base × height is still valid; use the trigonometric form when the height is not given.
• In an obtuse triangle, sin C stays positive, so the formula still gives a positive area.