Solve any triangle from any combination of sides and angles using the law of sines and cosines. Calculates all missing sides, angles, area, perimeter, and triangle type.
X is what % of Y: (X ÷ Y) × 100
X% of Y: (X ÷ 100) × Y
% change from A to B: ((B − A) ÷ A) × 100
Original value before X% added: Final ÷ (1 + X/100)
Original value before X% removed: Final ÷ (1 − X/100)
Worked examples: 45 is what % of 180? (45 ÷ 180) × 100 = 25%. 15% of 240? 0.15 × 240 = 36. Price rose from 80 to 96 — % increase: ((96−80) ÷ 80) × 100 = 20%. Price was 120 after 20% tax added — original pre-tax: 120 ÷ 1.20 = 100.
The law of sines states a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are side lengths and A, B, C are the opposite angles.
The law of cosines states c² = a² + b² − 2ab·cos(C). It generalises the Pythagorean theorem for any triangle.
These are triangle congruence cases: SSS = three sides known; SAS = two sides and included angle; ASA = two angles and included side; AAS = two angles and non-included side.
Use Heron's formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2, or use Area = ½ab·sin(C) if two sides and an angle are known.
An obtuse triangle has one angle greater than 90°. The law of cosines must be used to solve it, since the law of sines can give an ambiguous result.