How to solve a right triangle
Given the two legs a and b of a right triangle: the hypotenuse is √(a² + b²), the area is a × b ÷ 2, and the two acute angles are arctan(a ÷ b) and its complement. For legs 3 and 4: hypotenuse 5, area 6, angles 36.87° and 53.13°.
The area formula is the triangle half of a rectangle: two copies of the triangle placed together form a 3 by 4 rectangle of area 12, so one triangle covers 6. For the space inside rather than the sides themselves, use the triangle area calculator.
How to use this calculator
Enter the lengths of the two legs (the sides at the right angle). The result gives the hypotenuse, area and both non-right angles, which always sum to 90°. For just the hypotenuse, see the hypotenuse calculator.
Keep both legs in the same unit; the hypotenuse comes back in that unit and the area in square units. The angles never depend on scale — doubling both legs to 6 and 8 gives the same 36.87° and 53.13°.
Checking and using the result
The hypotenuse is always the longest side, yet always shorter than the two legs added together — which is why walking around a corner is longer than cutting across it.
The 3-4-5 set is the best known Pythagorean triple. Builders use it to square up a corner: measure 3 units along one edge and 4 along the other, and the diagonal reads exactly 5 when the angle is a true 90°. When you know both legs and want the angle between the hypotenuse and the base, the ratio you need is a tangent, handled by the tangent calculator.
Frequently asked questions
- How do I solve a right triangle from two legs?
- Hypotenuse = √(a² + b²), area = a × b ÷ 2, and the angles are arctan(a/b) and 90° minus that.
- What are the angles of a 3-4-5 triangle?
- About 36.87° and 53.13°, plus the 90° right angle.
- What is the area of a right triangle?
- Half the product of the two legs: area = a × b ÷ 2.
- Do the two acute angles always add to 90°?
- Yes. In a right triangle the non-right angles sum to 90°, since all three add to 180°.
- What if I know the hypotenuse and one leg instead?
- Find the missing leg first by rearranging the Pythagorean theorem: b = √(c² − a²). With a hypotenuse of 5 and a leg of 3, the other leg is √(25 − 9) = 4, and you can then enter both legs here.