Right Triangle Calculator
Provide any two of the following parameters to compute the remaining properties of a right‑angled triangle. When you choose radians for angles, you may enter values like π/3, π/4, and so on.
Right triangle
A right triangle is a type of triangle that has one angle that measures 90°. Right triangles, and the relationships between their sides and angles, are the basis of trigonometry.
In a right‑angled triangle the side opposite the 90° angle is the longest and is called the hypotenuse. We usually label the sides a, b and c, where c denotes the hypotenuse and a, b are the two shorter legs. The angles are denoted by the capital letters that match the opposite side: ∠A opposite side a, ∠B opposite side b, and ∠C (which equals 90°) opposite side c, as illustrated below. This tool uses the Greek letters α (alpha) and β (beta) for the unknown angles. The altitude h is the segment drawn from the right‑angle vertex down to the hypotenuse; it splits the original triangle into two smaller triangles that are each similar to the whole.
When all three sides of a right‑angled triangle are whole numbers the figure is known as a Pythagorean triangle. The three integer lengths are called a Pythagorean triple, for example 3‑4‑5, 5‑12‑13, 8‑15‑17, and so forth.
The area and perimeter of a right‑angled triangle are computed like any other triangle. The perimeter is simply the sum of the three side lengths, while the area can be obtained from the formula shown below.
| A = |
| ab = |
| ch |
Special Right Triangles
30°-60°-90° triangle:
A 30°‑60°‑90° triangle is a special right‑angled triangle whose angles are 30°, 60° and 90°. Its side lengths follow the proportion 1 : √3 : 2. Consequently, if you know the length of one side together with its opposite angle, the remaining sides can be derived from this ratio. For instance, if the side opposite the 60° angle measures 5, let a be the side opposite 30°, b the side opposite 60° (which is 5), and c the side opposite 90°.
Angles: 30°: 60°: 90°
Ratio of sides: 1:√3:2
Side lengths: a:5:c
Then using the known ratios of the sides of this special type of triangle:
| a = |
| = |
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| c = |
| = |
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Thus, knowing just a single side of a 30°‑60°‑90° triangle is enough to calculate the other two lengths quickly. This triangle is also handy for evaluating trigonometric values at multiples of π/6.
45°-45°-90° triangle:
The 45°-45°-90° triangle, also referred to as an isosceles right triangle, since it has two sides of equal lengths, is a right triangle in which the sides corresponding to the angles, 45°-45°-90°, follow a ratio of 1:1:√2. Like the 30°-60°-90° triangle, knowing one side length allows you to determine the lengths of the other sides of a 45°-45°-90° triangle.
Angles: 45°: 45°: 90°
Ratio of sides: 1:1:√2
Side lengths: a:a:c
Given c= 5:
| a = |
| = |
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45°-45°-90° triangles can be used to evaluate trigonometric functions for multiples of π/4.