Right Triangle Calculator — Solve Any Right Triangle
Enter any two known values of a right triangle and calculate all remaining sides, angles, area, perimeter, and altitude to the hypotenuse. See also Trigonometry Calculator and Pythagorean Theorem Calculator.
How to Solve a Right Triangle
A right triangle has one 90° angle and two acute angles that sum to 90°. To solve it completely, you need at least two known values (besides the right angle). Enter two sides, a side and an angle, or the hypotenuse with a side or angle. The calculator uses the Pythagorean theorem (a² + b² = c²) and trigonometric ratios (sin, cos, tan) to find all missing values including area, perimeter, and the altitude from the right angle to the hypotenuse.
Right Triangle Formulas
Pythagorean theorem: a² + b² = c²
Angle A = arctan(a / b) or arcsin(a / c)
Angle B = 90° − Angle A
Area = ½ × a × b
Perimeter = a + b + c
Altitude to hypotenuse h = (a × b) / c
sin A = a / c, cos A = b / c, tan A = a / b
Example: The Classic 3-4-5 Triangle
Given: a = 3, b = 4
c = √(3² + 4²) = √(9 + 16) = √25 = 5
Angle A = arctan(3/4) ≈ 36.87°
Angle B = 90° − 36.87° ≈ 53.13°
Area = ½ × 3 × 4 = 6
Perimeter = 3 + 4 + 5 = 12
Altitude to hypotenuse = (3 × 4) / 5 = 2.4
Common Right Triangles Reference
| a | b | c | Angle A | Angle B | Area |
|---|---|---|---|---|---|
| 3 | 4 | 5 | 36.87° | 53.13° | 6 |
| 5 | 12 | 13 | 22.62° | 67.38° | 30 |
| 8 | 15 | 17 | 28.07° | 61.93° | 60 |
| 7 | 24 | 25 | 16.26° | 73.74° | 84 |
| 1 | 1 | 1.4142 | 45° | 45° | 0.5 |
| 1 | 1.7321 | 2 | 30° | 60° | 0.87 |
Frequently Asked Questions
What is a right triangle?
A right triangle is a triangle with one angle exactly equal to 90°. The side opposite the right angle is called the hypotenuse and is always the longest side. The other two sides are called legs (a and b).
How many values do I need to solve a right triangle?
You need exactly two known values besides the right angle. This can be two sides, one side and one acute angle, or the hypotenuse and one leg. With any two values, all remaining sides and angles can be calculated.
What is the altitude to the hypotenuse?
The altitude to the hypotenuse is the perpendicular line drawn from the right angle vertex to the hypotenuse. Its length equals (a × b) / c. It divides the hypotenuse into two segments and creates two smaller triangles that are similar to the original.
What are special right triangles?
The two most common special right triangles are the 45-45-90 triangle (isosceles, sides in ratio 1:1:√2) and the 30-60-90 triangle (sides in ratio 1:√3:2). These appear frequently in geometry, trigonometry, and standardized tests.
Can a right triangle have two equal sides?
Yes — a 45-45-90 triangle has two equal legs. It is both a right triangle and an isosceles triangle. The hypotenuse is √2 times the length of each leg. However, a right triangle cannot be equilateral since all angles in an equilateral triangle are 60°.