Solve a right triangle from any two values, with angles, area and perimeter.
A right triangle has one 90° angle. Given any two sides (or a side and an acute angle), all remaining values follow from the Pythagorean theorem (a² + b² = c²) and the six trigonometric functions. Angles A and B are always complementary: A + B = 90°. The area is simply half the product of the two legs.
Pythagorean theorem
c = √(a² + b²)
Angles from sides
A = arctan(a/b), B = 90° − A
Area
Area = a × b / 2
It states that for any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². The classic example is the 3-4-5 triangle: 3² + 4² = 9 + 16 = 25 = 5².
Use trigonometry. If you know leg a and angle A (opposite to a), then b = a / tan(A) and c = a / sin(A). If you know the hypotenuse and an angle, the legs are c·sin(A) and c·cos(A).
The right angle is exactly 90°, and the remaining two angles A and B must together fill the other 90°, so A + B + 90° = 180°. This is true for every triangle — the sum of interior angles is always 180°.
The two shorter sides that form the right angle are called legs (a and b). The side opposite the right angle — always the longest side — is the hypotenuse (c).