Right Triangle Calculator
Solve a right triangle from any two measurements — two legs, a leg and the hypotenuse, or a side and an angle. Every result comes with the working, so you can check the method as well as the answer.
Sides and angles
- Leg a
- 3
- Leg b
- 4
- Angle A
- 36.87°
- Angle B
- 53.13°
- Area
- 6
- Perimeter
- 12
Derived measures
- Altitude to c
- 2.4
- Inradius
- 1
- Circumradius
- 2.5
Working
- c = √(a² + b²) = √(3² + 4²) = 5
- A = arcsin(a / c) = arcsin(3 / 5) = 36.8699°
- B = 90° − A = 53.1301°
- Area = (a × b) / 2 = (3 × 4) / 2 = 6
Primitive Pythagorean triples
Right triangles whose three sides are whole numbers with no common factor. Every one can be generated by Euclid's formula(m² − n², 2mn, m² + n²) for coprime integers m > n of opposite parity. Any multiple of a triple is also a triple.
How a right triangle is solved
From two sides
c = √(a² + b²) given both legs b = √(c² − a²) given a leg and the hypotenuse
From a side and an angle
a = c · sin(A) b = c · cos(A) c = a / sin(A) b = a / tan(A) c = b / cos(A) a = b · tan(A)
Which formula applies depends on whether the side you have is opposite or adjacent to the angle you have. Angle A is opposite leg a throughout this page, which is the standard labelling convention.
Derived quantities
Area = (a · b) / 2 Altitude = (a · b) / c perpendicular from the right angle to c Inradius = (a + b − c) / 2 simplifies from Area / s for a right triangle Circumradius = c / 2 the hypotenuse is a diameter (Thales)
The circumradius result is Thales' theorem in disguise: any triangle inscribed in a circle with one side as the diameter is right-angled, so the hypotenuse of a right triangle is always a diameter of its circumscribed circle and the circumcentre sits at its midpoint.
Frequently asked questions
What is the minimum I need to solve a right triangle?
Any two independent measurements, as long as they are not both acute angles. The right angle is already known, so two legs, a leg and the hypotenuse, or any side with either acute angle all determine the triangle completely. Two acute angles fix the shape but not the size, so they produce a family of similar triangles rather than one answer.
What is the Pythagorean theorem?
In a right triangle, a² + b² = c², where c is the hypotenuse — the side opposite the right angle, and always the longest. Rearranged, a missing leg is √(c² − a²). It applies only to right triangles; the law of cosines, c² = a² + b² − 2ab·cos(C), generalises it to any triangle and reduces to Pythagoras when C is 90° because cos(90°) = 0.
How do I find the angles of a right triangle?
Use inverse trigonometric functions: angle A = arcsin(a / c), or arccos(b / c), or arctan(a / b) — whichever ratio you have. The two acute angles always sum to 90°, since the interior angles of any triangle total 180° and one is already taken by the right angle, so finding one immediately gives the other by subtraction.
What is SOH-CAH-TOA?
A mnemonic for the three basic ratios: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. "Opposite" and "adjacent" are relative to the angle you are working with, which is the part that trips people up — swapping the angle swaps which leg is which.
What are the special right triangles?
Two recur constantly. The 45-45-90 triangle has sides in the ratio 1 : 1 : √2, so the hypotenuse is always a leg times √2. The 30-60-90 triangle has sides 1 : √3 : 2, with the shortest side opposite the 30° angle and exactly half the hypotenuse. Recognising them lets you solve by inspection rather than with a calculator.
What is a Pythagorean triple?
Three whole numbers satisfying a² + b² = c², such as 3-4-5, 5-12-13 and 8-15-17. Any multiple of a triple is also a triple, so 6-8-10 works too; triples that share no common factor are called primitive. Builders use the 3-4-5 relationship to square up corners without a set square — measure 3 units along one edge and 4 along the other, and the diagonal reads exactly 5 when the corner is true.
How do I find the area without knowing the height?
In a right triangle the two legs are perpendicular, so they serve directly as base and height: area = (a × b) / 2. No separate height measurement is needed. If you only have the hypotenuse and one leg, find the other leg with the Pythagorean theorem first. The altitude to the hypotenuse — a different quantity — is (a × b) / c.
Why does the calculator reject some inputs?
Because they describe an impossible triangle. The hypotenuse must be strictly longer than either leg, so entering a leg of 6 with a hypotenuse of 5 has no solution. Likewise an acute angle must be under 90°, since the triangle already contains a right angle and the three interior angles must total 180°.