Solve any right triangle from 2 known values. Enter two sides or one side + one acute angle to compute all sides, both angles, area, perimeter, altitude to hypotenuse, inradius, circumradius, and P...
A right triangle is a triangle with one 90° angle. It is arguably the most important shape in all of mathematics and science. The Pythagorean theorem — a² + b² = c², where c is the hypotenuse — is one of the oldest and most widely used mathematical formulas in human history.
Solving a right triangle means finding all unknown sides and angles from the known values. You need at least two pieces of information (beyond the right angle itself). Common input combinations include two legs (a and b), a leg and the hypotenuse (a and c, or b and c), or one side and one acute angle.
Beyond sides and angles, right triangles have rich geometric properties. The altitude from the right angle to the hypotenuse creates two smaller triangles, each similar to the original. The inradius r = (a + b − c)/2 gives the radius of the largest inscribed circle. The circumradius R = c/2 — uniquely simple because the hypotenuse is always a diameter of the circumscribed circle (Thales' theorem).
Pythagorean triples — integer solutions to a² + b² = c² — have fascinated mathematicians for millennia. The most famous is (3, 4, 5). A primitive triple is one where the three integers share no common factor greater than 1.
This calculator handles all input combinations, computes every property at once, checks for Pythagorean triples, and displays a reference table of common triples for exploration.
This solver is useful when you need more than just the missing hypotenuse. Depending on what you know, it can start from two sides or from one side with one acute angle, then return every major geometric property of the same triangle in one pass. That saves time when you are checking homework, comparing multiple design options, or confirming measurements in construction and drafting.
Because the calculator also reports altitude to the hypotenuse, inradius, circumradius, and Pythagorean triple status, it works well as both a classroom study tool and a practical field reference. Instead of switching between separate formulas for side recovery, angle finding, and area, you can verify the full triangle consistently from one set of inputs.
Pythagorean theorem: a² + b² = c² Area: A = ½ab Perimeter: P = a + b + c Altitude to hypotenuse: h = ab/c Inradius: r = (a + b − c)/2 Circumradius: R = c/2 Angles: A = arctan(a/b), B = 90° − A
Result: Hypotenuse = 5, Area = 6, Perimeter = 12, Pythagorean Triple: Yes (Primitive)
With legs 3 and 4: hypotenuse = √(9+16) = √25 = 5. Area = ½×3×4 = 6. Perimeter = 3+4+5 = 12. Altitude to hypotenuse = 3×4/5 = 2.4. Inradius = (3+4−5)/2 = 1. Circumradius = 5/2 = 2.5. (3,4,5) is a primitive Pythagorean triple.
A right triangle is determined once you know any two independent measurements beyond the fixed 90° angle. In practice, that usually means two sides or one side and one acute angle. This calculator is designed around those exact cases, so you can switch between leg-leg, leg-hypotenuse, or side-angle combinations without having to rethink which formula comes next. That is especially helpful when textbook problems present the same triangle in different ways.
Many right-triangle tools stop after reporting the missing side lengths, but the secondary measurements are often the most useful ones. The altitude to the hypotenuse appears in similarity proofs, the inradius helps with inscribed-circle questions, and the circumradius matters whenever the triangle is tied to a circle. Seeing those values together makes it easier to spot relationships such as $R = c/2$ and $h = ab/c$ without recomputing each one by hand.
Full right-triangle solving shows up in carpentry, roof layout, ramps, navigation, and coordinate geometry. A builder may know a rise and run, a student may know one leg and an acute angle, and a survey problem may give a distance with a bearing-based angle. In all of those cases, the goal is the same: recover the complete triangle accurately and check whether the numbers behave like a known Pythagorean triple before using them elsewhere.
Use the Pythagorean theorem: c = √(a² + b²), where a and b are the two legs. For example, legs 3 and 4 give hypotenuse √(9+16) = √25 = 5.
Rearrange the Pythagorean theorem: a = √(c² − b²). For example, hypotenuse 13 and leg 5 give the other leg √(169−25) = √144 = 12.
A Pythagorean triple is a set of three positive integers (a, b, c) where a² + b² = c². Common examples: (3,4,5), (5,12,13), (8,15,17), (7,24,25).
It is the perpendicular distance from the right angle vertex to the hypotenuse. h = ab/c. It divides the triangle into two smaller similar triangles.
By Thales' theorem, any triangle inscribed in a semicircle with the hypotenuse as diameter is a right triangle. Therefore the hypotenuse is the diameter and R = c/2.
Angle A (opposite leg a) = arctan(a/b) or arcsin(a/c). Angle B = 90° − A. For legs 3 and 4: A = arctan(3/4) ≈ 36.87°, B ≈ 53.13°.