Right Triangle Calculator
This comprehensive calculator can solve a right-angled triangle from any two known values. Enter your values to find all missing sides, angles, area, and perimeter.
Your Results
Select the values you know, enter them, and click "Solve Triangle".
A right triangle with sides a, b, hypotenuse c, and corresponding opposite angles A (α) and B (β).
Solving the Right-Angled Triangle
A right-angled triangle (or right triangle) is a triangle in which one angle is exactly 90 degrees. This special property allows us to "solve" the triangle—that is, find all unknown sides and angles—using a minimum of two known pieces of information, as long as at least one is a side.
The sides adjacent to the right angle are called legs (sides a and b), and the side opposite the right angle is the hypotenuse (side c), which is always the longest side.
Core Formulas Used
Solving a right triangle relies on two key mathematical principles:
1. The Pythagorean Theorem
When two sides are known, the third can be found using the Pythagorean theorem. This theorem is a cornerstone of geometry and has its own dedicated tool, our Pythagorean Theorem Calculator.
2. Trigonometric Functions (SOH-CAH-TOA)
When one side and one angle are known, we use trigonometric functions to find the missing parts. The mnemonic SOH-CAH-TOA helps remember the ratios:
- SOH: Sin(angle) = Opposite / Hypotenuse
- CAH: Cos(angle) = Adjacent / Hypotenuse
- TOA: Tan(angle) = Opposite / Adjacent
For example, `sin(A) = a/c` and `tan(B) = b/a`.
Other Key Properties
- Sum of Angles: The three angles in any triangle always add up to 180°. Since one angle is 90°, the other two acute angles (A and B) must add up to 90°. (A + B = 90°).
- Area: The area of a right triangle is half the product of its two legs. Area = ½ × a × b. You can find area formulas for other shapes with our Area Calculator.
- Perimeter: The perimeter is the sum of the lengths of all three sides. Perimeter = a + b + c.
This calculator is an invaluable tool for students in trigonometry, as well as for professionals in fields like construction, engineering, and physics. For more advanced math tools, please visit our main Math calculators page.