Right Triangle Calculator
Calculate all properties of a right triangle. Enter any two side lengths to find the hypotenuse, angles, area, perimeter, and all three altitudes.
Right Triangle Properties
Common Pythagorean Triples
Click any triple to load it into the calculator with mode set to sides a and b.
| Property | Formula |
|---|---|
| Hypotenuse c | c = √(a² + b²) |
| Leg b | b = √(c² − a²) |
| Perimeter P | P = a + b + c |
| Semiperimeter s | s = (a + b + c) / 2 |
| Area K | K = (a × b) / 2 |
| Altitude hₐ | hₐ = b |
| Altitude h_b | h_b = a |
| Altitude h_c | h_c = (a × b) / c |
| Angle A | A = arctan(a / b) |
| Angle B | B = arctan(b / a) |
Right Triangle Formulas Explained
A right triangle is any triangle containing one 90-degree angle. This single constraint gives rise to a complete set of predictable relationships between the sides and angles, all rooted in the Pythagorean Theorem.
The Pythagorean Theorem
The Pythagorean Theorem states that a² + b² = c², where a and b are the two legs (the sides that form the right angle) and c is the hypotenuse (the side opposite the right angle). This relationship is the foundation of all right triangle calculations.
Angles in a Right Triangle
Because one angle is always 90°, the other two angles must sum to 90°. These are called complementary angles. If you know any one of them, you immediately know the other. Angle A is found with arctan(a/b) and Angle B = 90° minus Angle A.
Altitudes of a Right Triangle
Each altitude is the perpendicular distance from a vertex to the opposite side. In a right triangle, the altitude from the right-angle vertex to the hypotenuse equals (a × b) / c. The altitudes to the two legs are simply the other leg: hₐ = b and h_b = a.
Pythagorean Triples Reference
Pythagorean triples are sets of three whole numbers that perfectly satisfy a² + b² = c². They are commonly used in construction, carpentry, and navigation to create precise right angles without measuring instruments.
Generating Odd Pythagorean Triples
For any odd number n greater than 1, a triple can be generated: a = n, b = (n² minus 1) / 2, c = (n² plus 1) / 2. For example, with n = 5: b = 12, c = 13, giving the triple (5, 12, 13).
Generating Even Pythagorean Triples
For any even number n, the triple is: a = n, b = (n/2)² minus 1, c = (n/2)² plus 1. For example, with n = 8: b = 15, c = 17, giving the triple (8, 15, 17).
Scaled Pythagorean Triples
Any Pythagorean triple can be scaled by multiplying all three values by the same integer. For example, (3, 4, 5) scaled by 2 gives (6, 8, 10), and scaled by 3 gives (9, 12, 15). All scaled versions are valid right triangles.
Frequently Asked Questions
Use the Pythagorean Theorem: c = √(a² + b²), where a and b are the two legs and c is the hypotenuse. Square both legs, add them together, then take the square root of the result.
The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It applies only to triangles that contain one 90-degree angle.
The area of a right triangle is half the product of the two legs: K = (a × b) / 2. The two legs are the sides that meet at the right angle, not the hypotenuse.
Pythagorean triples are sets of three positive whole numbers that satisfy a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). They represent right triangles with integer side lengths.
A triangle has three altitudes, each perpendicular to one side. For a right triangle: altitude to side a equals b, altitude to side b equals a, and altitude to the hypotenuse c equals (a × b) / c.
Angle C is always 90°. Angle A = arctan(a / b) and Angle B = 90° minus Angle A. Alternatively, use arcsin or arccos with the known sides and hypotenuse.