How the right triangle calculator works
A right triangle has one 90-degree angle. The side opposite that angle is the hypotenuse c, and it is always the longest side. The two shorter sides are legs a and b.
Two independent measurements are enough to solve a right triangle when at least one measurement establishes its size. The calculator applies the Pythagorean theorem and trigonometric relationships to determine the missing values.
Right triangle formulas
- Pythagorean theorem: a² + b² = c²
- Acute angles: α + β = 90°
- Area: A = ab ÷ 2
- Perimeter: P = a + b + c
- Altitude to the hypotenuse: h = ab ÷ c
Special right triangles
A 45°–45°–90° triangle has equal legs in the ratio 1:1:√2. A 30°–60°–90° triangle has sides in the ratio 1:√3:2. Integer side sets such as 3–4–5 and 5–12–13 are called Pythagorean triples.
Choosing valid known values
You can enter combinations such as two sides, one side and one acute angle, a side and area, or hypotenuse and perimeter. Two acute angles are not enough because they fix the proportions without defining the triangle’s size.
Some combinations that do not distinguish the two legs can describe the same triangle reflected across its altitude. In that situation, the calculator reports the orientation with a ≤ b.