What is a Right Triangle?
A right triangle (or right-angled triangle) is a specific type of triangle that has exactly one angle measuring 90° (a right angle). Right triangles, and the relationships between their sides and angles, form the foundational basis of trigonometry.
In a right triangle, the side directly opposite the 90° angle is always the longest side of the triangle, and it is known as the hypotenuse. The other two shorter sides are often referred to as the legs of the triangle. The relationships between these lengths are best defined by the Pythagorean Theorem.
In common geometric notation:
- c refers to the hypotenuse.
- a and b represent the lengths of the two shorter legs.
- Angle A (or α) is the angle directly opposite side a.
- Angle B (or β) is the angle directly opposite side b.
- h represents the altitude (the perpendicular length from the right angle vertex to the hypotenuse).
How to Use This Calculator
This calculator can solve the entire right triangle as long as you provide exactly two known values. These values can be a combination of two sides, one side and one angle, the area and a side, or the perimeter and a side.
Simply enter your known values into the corresponding input fields. If you are using radians, you can input algebraic expressions such as pi/3 or pi/4. The calculator will automatically perform the necessary trigonometric and algebraic calculations to find all missing values, including area and perimeter.
Right Triangle Equations and Formulas
The calculations are performed using standard geometric formulas and trigonometric ratios. If all three sides of a right triangle are integers, they form a "Pythagorean triple" (e.g., 3-4-5 or 5-12-13).
Area (A): ½ × a × b = ½ × c × h
Perimeter (P): a + b + c
Altitude (h): (a × b) / c
The trigonometric ratios for a right triangle are:
Cosine: cos(α) = b / c
Tangent: tan(α) = a / b
Special Right Triangles
30°-60°-90° Triangle
This special triangle refers to the angle measurements in degrees. In this type of right triangle, the sides corresponding to the angles 30°-60°-90° follow a specific, predictable ratio of 1 : √3 : 2. Knowing just one side of a 30°-60°-90° triangle allows you to determine the lengths of the other sides relatively easily. It is frequently used to evaluate trigonometric functions for multiples of π/6.
45°-45°-90° Triangle
Also known as an isosceles right triangle, this triangle features two equal angles of 45° and two legs of equal length. The sides corresponding to the angles 45°-45°-90° follow a fixed ratio of 1 : 1 : √2. Like the previous example, knowing one side length lets you quickly scale the ratio to find the rest. It is commonly used to evaluate trigonometric functions for multiples of π/4.
Frequently Asked Questions
If you only know the angles (such as 90°, 30°, and 60°), you cannot determine the exact lengths of the sides. The angles only tell you the shape (ratio) of the triangle, not its physical size. You need at least one unit of length (a side, area, or perimeter) to calculate exact dimensions.
First, ensure the "Radians (rad)" toggle is selected at the top of the calculator. You can then simply type "pi/4" or "pi/3" directly into the angle input fields. The calculator will automatically convert it into a numerical value for the calculation.
The altitude (h) is the length of a perpendicular line drawn from the vertex of the 90° angle directly to the hypotenuse (the longest side). Drawing this altitude actually divides the original triangle into two smaller, similar right triangles.
The calculator requires mathematically valid inputs to form a real right triangle. For example, if you enter a hypotenuse (c) that is smaller than one of the legs (a or b), it is impossible to construct such a triangle, and the calculator will return an error. Similarly, the two acute angles must add up exactly to 90°.
A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². They describe right triangles where all three side lengths are perfectly whole numbers. Common examples include 3-4-5, 5-12-13, and 8-15-17.