The formula
How it works
Enter the two legs of a right-angled triangle and this calculator finds everything else: the hypotenuse, the area, the perimeter and the two remaining angles. It combines the Pythagorean theorem with basic trigonometry.
FAQ
What makes a triangle “right”?
A right triangle has one angle of exactly 90 degrees. The two sides that form that angle are the legs, and the longest side opposite it is the hypotenuse.
How are the angles found?
From the legs using trigonometry: the angle opposite leg a is arctan(a ÷ b). The two non-right angles always add up to 90 degrees, so the third follows immediately.
Can I enter the hypotenuse instead of a leg?
This calculator expects the two legs, not the hypotenuse. If you only know one leg and the hypotenuse, find the missing leg first with \(b = \sqrt{c^2 - a^2}\), then enter both legs here.
Does it matter which leg I call a and which I call b?
No — swapping a and b just swaps which acute angle is labelled α and which is β. The hypotenuse, area and perimeter come out identical either way.
What are the 3-4-5 and 45-45-90 triangles?
They are well-known right triangles used as quick checks: legs of 3 and 4 always give a hypotenuse of 5, and equal legs always give two 45-degree angles. Entering equal values for a and b reproduces the 45-45-90 case.
Can the legs be decimals or very large numbers?
Yes, the calculator accepts any positive decimal value for each leg and scales the hypotenuse, area, perimeter and angles accordingly. Just keep both legs in the same unit so the results stay consistent.
Why is the area just half of a times b?
In a right triangle the two legs are perpendicular, so one can be treated as the base and the other as the height. The general triangle area formula, half of base times height, then simplifies directly to \(\tfrac{1}{2}ab\).
About the right triangle calculator
This calculator solves a right-angled triangle completely from just its two legs. A right triangle is the most useful shape in geometry and trigonometry, because its 90-degree angle links its sides and angles through simple, exact rules. Give it the two shorter sides and it returns the hypotenuse, the area, the perimeter and both of the other angles, saving you from juggling several formulas by hand.
How to use it
Enter the lengths of the two legs — the sides that meet at the right angle. The calculator returns the hypotenuse, the area, the perimeter, and the two acute angles. For example, legs of 3 and 4 give the classic 3-4-5 triangle: a hypotenuse of 5, an area of 6, a perimeter of 12, and angles of about 36.9 and 53.1 degrees. Use any units you like — the sides come back in the same unit and the area in that unit squared.
The formula
The hypotenuse comes from the Pythagorean theorem, . The area of a right triangle is half the product of its legs, , because the legs act as the base and height. The angles use trigonometry: the angle opposite leg is , and since the angles of any triangle sum to 180 degrees, the other acute angle is whatever is left after taking away the 90-degree angle.
Where it is used
Right triangles are the workhorses of building, surveying and navigation. Carpenters use the leg-and-hypotenuse relationship to cut braces and rafters, and the 3-4-5 rule to check that corners are square. Surveyors and sailors use the angles to measure distances and bearings indirectly, and the same triangle underlies the trigonometry behind ramps, roofs, staircases and the coordinate geometry used in computer graphics.