Enter any two known values (sides, angles, height, area, or perimeter) and the calculator finds everything else. Sides are a and b (the legs) and c (the hypotenuse); ∠α is opposite a and ∠β is opposite b.
A right triangle calculator takes the guesswork out of trigonometry. Enter any two known measurements - two sides, a side and an angle, a side and the area, and many other combinations - and this tool instantly computes every remaining property of the triangle: the third side, both acute angles, the height to the hypotenuse, the area, the perimeter, the inradius, and the circumradius. It even draws the triangle to scale and shows the formulas behind each step. Whether you are a student checking homework, a carpenter laying out a square corner, or an engineer working through a design, this calculator gives fast, accurate answers.
A right triangle (or right-angled triangle) is a triangle in which one of the three interior angles is exactly 90 degrees - a right angle. That square corner is the defining feature, and it gives right triangles a set of special properties that make them the foundation of trigonometry, surveying, navigation, construction, and physics.
The side opposite the right angle is the longest side and is called the hypotenuse, labeled c on this calculator. The other two sides, which form the right angle, are called the legs and are labeled a and b. The two non-right angles, ∠α (alpha) and ∠β (beta), are always acute (less than 90 degrees) and always add up to 90 degrees, because the three angles of any triangle sum to 180 degrees and the right angle already uses 90 of them. On this page, ∠α is the angle opposite side a and ∠β is the angle opposite side b.
The single most important relationship in any right triangle is the Pythagorean theorem, which connects the three sides:
a2 + b2 = c2
In words, the square of the hypotenuse equals the sum of the squares of the two legs. This lets you find any side when you know the other two. If a = 3 and b = 4, then c = √(3² + 4²) = √25 = 5. If you know the hypotenuse and one leg, rearrange it: b = √(c² − a²). The famous 3-4-5 triangle - the default on this calculator - is the simplest example of a right triangle with whole-number sides.
The theorem only works for right triangles, which is exactly why it is so useful for testing whether a corner is truly square: measure 3 units along one edge, 4 units along the other, and if the diagonal between those points is exactly 5 units, the corner is a perfect right angle. Builders have used this "3-4-5 rule" for thousands of years.
The angles of a right triangle are linked to its sides through the three primary trigonometric ratios. For an acute angle, with sides described relative to that angle as the opposite, the adjacent, and the hypotenuse:
For angle α in our triangle, the opposite side is a and the adjacent side is b, so sinα = a/c, cosα = b/c, and tanα = a/b. To recover an angle from a ratio you use the inverse functions (arcsine, arccosine, arctangent). For instance, α = arctan(a/b). This is how the calculator turns side lengths into angles, and angles back into side lengths.
Because the two legs of a right triangle are perpendicular, they serve directly as the base and height, making the area especially easy to compute:
Area = ½ × a × b
For the 3-4-5 triangle, the area is ½ × 3 × 4 = 6 square units. If you do not know both legs but you do know one leg and an angle, the calculator first finds the missing leg using a trigonometric ratio, then applies the area formula. If you know the area and one leg, it reverses the process to find the other leg.
The perimeter is simply the sum of all three sides, P = a + b + c. The height (h) reported by this calculator is the altitude drawn from the right angle perpendicular to the hypotenuse, given by h = (a × b) ÷ c. This altitude divides the right triangle into two smaller triangles that are each similar to the original - a fact used throughout geometry.
The inradius (r) is the radius of the largest circle that fits inside the triangle, touching all three sides. For a right triangle there is an elegant shortcut: r = (a + b − c) ÷ 2. The circumradius (R) is the radius of the circle that passes through all three vertices. Remarkably, for a right triangle the hypotenuse is always a diameter of that circle, so R = c ÷ 2. This is a direct consequence of Thales' theorem.
You only need two independent pieces of information to pin down a right triangle (the 90-degree angle counts as already known). The calculator accepts many combinations, including:
Certain combinations - notably the hypotenuse paired with the area, the hypotenuse paired with the perimeter, or the area paired with the perimeter - can describe two different right triangles, because the two legs can swap roles. When that happens, the calculator reports both possible solutions.
Two right triangles appear so often that their side ratios are worth memorizing.
An isosceles right triangle has two 45-degree angles and two equal legs. Its sides are always in the ratio 1 : 1 : √2. If each leg has length x, the hypotenuse is x√2. This triangle is exactly half of a square cut along its diagonal, which is why it shows up constantly in design and construction.
This triangle has angles of 30, 60, and 90 degrees, with sides in the ratio 1 : √3 : 2. The side opposite the 30-degree angle is the shortest, the side opposite the 60-degree angle is √3 times longer, and the hypotenuse is twice the shortest side. It is exactly half of an equilateral triangle, and it underpins many exact trigonometric values.
A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c², producing a right triangle with integer sides. The best known is 3-4-5, but there are infinitely many, including 5-12-13, 8-15-17, 7-24-25, and 20-21-29. Any multiple of a triple is also a triple, so 6-8-10 and 9-12-15 are simply scaled-up 3-4-5 triangles. These triples are prized in construction and design because they let you build exact right angles using only a tape measure, with no protractor required.
Right triangles are everywhere once you start looking:
Two. Because one angle is already fixed at 90 degrees, any two additional independent measurements are enough to determine the whole triangle. At least one of them should be a length so the triangle has a definite size - two angles alone fix only the shape.
The hypotenuse is the side opposite the right angle, and it is always the longest side. On this calculator it is labeled c, while a and b are the two shorter sides (the legs) that meet at the right angle.
When you supply values that fix the size but not which leg is which - for example the hypotenuse and the area - the two legs can trade places, producing two valid right triangles that are mirror images. The calculator shows both so nothing is missed.
Yes. Each angle box has a dropdown to switch between degrees and radians. Results display angles in decimal degrees, in degrees-minutes-seconds, and in radians, so you can use whichever format you need.
It is the altitude from the right angle to the hypotenuse, calculated as h = (a × b) ÷ c. This is different from the legs themselves, which can also be thought of as heights relative to each other.
No. All calculations happen in your browser. Nothing you type is uploaded or saved to any server, so your data stays private.
This Right Triangle Calculator is provided for educational and general informational purposes. It uses standard geometric and trigonometric formulas and is suitable for schoolwork, DIY projects, and everyday calculations. For professional construction or engineering work, always verify critical measurements independently.