Right Triangle Calculator - Find Missing Sides & Angles

The Pythagorean theorem

For any right triangle with legs a and b and hypotenuse c: a² + b² = c². Enter any two of the five values and the calculator instantly finds the remaining three.

Trigonometric ratios (SOH-CAH-TOA)

sin(A) = a / c (Opposite over Hypotenuse). cos(A) = b / c (Adjacent over Hypotenuse). tan(A) = a / b (Opposite over Adjacent). These identities let the calculator find side lengths when an angle is given.

Common right-triangle ratios

Well-known Pythagorean triples include 3-4-5, 5-12-13, 8-15-17, and 7-24-25. The 45-45-90 triangle has legs in ratio 1 : 1 : √2, and the 30-60-90 triangle has sides in ratio 1 : √3 : 2.

Worked example

Given angle A = 30° and hypotenuse c = 10, find all sides and the remaining angle:

1. Angle B = 90° − 30° = 60°
2. Side a (opposite A) = c × sin(30°) = 10 × 0.5 = 5
3. Side b (adjacent to A) = c × cos(30°) = 10 × (√3/2) ≈ 8.66
4. Check: a² + b² = 25 + 75 = 100 = c² ✓

Special right triangles

• 45–45–90: both legs equal; hypotenuse = leg × √2. Arises from a square bisected diagonally. Example: leg = 7 → hypotenuse = 7√2 ≈ 9.90.
• 30–60–90: sides in ratio 1 : √3 : 2. Arises from an equilateral triangle bisected by an altitude. Example: shortest side = 5 → longer leg = 5√3 ≈ 8.66, hypotenuse = 10.

Practical applications

• Staircase pitch: a staircase with a 7-inch rise and 11-inch run forms a right triangle; the stringer length (hypotenuse) = √(7² + 11²) ≈ 13 inches per step.
• Tree height from shadow: if a tree casts a 20 ft shadow and the sun angle is 35°, height = 20 × tan(35°) ≈ 14 ft.
• Navigation triangulation: dead reckoning uses right triangles to calculate position; a ship that travels 50 miles east then 30 miles north is √(50² + 30²) ≈ 58.3 miles from its starting point.