Right Triangle Calculator
Solve a right triangle from any two known values — sides or angles — and get all angles, area and perimeter.
Inputs
Hypotenuse c
5.000000
Leg a
3.000000
Leg b
4.000000
Angle A
36.8699°
Angle B
53.1301°
Area
6.000000
Perimeter
12.000000
Step by step
Apply Pythagorean theorem
a² + b² = c² → 3² + 4² = 5²
= c = √(9 + 16) = 5
Angle A (opposite a)
arctan(a/b) = arctan(3/4)
= 36.869898°
Angle B (opposite b)
90° − A = 90° − 36.869898°
= 53.130102°
Area
½ × a × b = ½ × 3 × 4
= 6
Perimeter
a + b + c = 3 + 4 + 5
= 12
How it works
A right triangle has one 90° angle. Given any two sides (or a side and an acute angle), all remaining values follow from the Pythagorean theorem (a² + b² = c²) and the six trigonometric functions. Angles A and B are always complementary: A + B = 90°. The area is simply half the product of the two legs.
Formulas
Pythagorean theorem
c = √(a² + b²)
- a
- Leg a (shorter side)
- b
- Leg b
- c
- Hypotenuse
Angles from sides
A = arctan(a/b), B = 90° − A
Area
Area = a × b / 2
Frequently Asked Questions
What is the Pythagorean theorem?
It states that for any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². The classic example is the 3-4-5 triangle: 3² + 4² = 9 + 16 = 25 = 5².
How do you find a missing side when you have a side and an angle?
Use trigonometry. If you know leg a and angle A (opposite to a), then b = a / tan(A) and c = a / sin(A). If you know the hypotenuse and an angle, the legs are c·sin(A) and c·cos(A).
Why do right triangle angles always add to 180°?
The right angle is exactly 90°, and the remaining two angles A and B must together fill the other 90°, so A + B + 90° = 180°. This is true for every triangle — the sum of interior angles is always 180°.
What is the difference between legs and the hypotenuse?
The two shorter sides that form the right angle are called legs (a and b). The side opposite the right angle — always the longest side — is the hypotenuse (c).