Pythagoras Triangle Calculator

Enter any two sides of a right triangle (leave the unknown one blank) to find the missing side, area, perimeter, and interior angles using the Pythagorean theorem, a² + b² = c².

Quick Facts

Pythagorean theorem
a² + b² = c²
c is the hypotenuse, the longest side, opposite the right angle.
Area formula
Area = ½ × a × b
The two legs are perpendicular, so one is the base and the other the height.
Angle formula
A = sin⁻¹(a / c)
The two non-right angles always add up to 90°.

Your Results

Calculated
Solved Side
-
Missing side from a² + b² = c²
Area
-
½ × leg a × leg b
Perimeter
-
a + b + c
Angles (opposite a / b)
-
The right angle is 90°

Ready

Enter two sides (leave the unknown side blank) and press Calculate.

Formula and Method for the Pythagoras Triangle Calculator

A right triangle has one 90° angle. The side opposite that right angle, called the hypotenuse (c), is always the longest side. The two remaining sides, called legs (a and b), meet at the right angle. The Pythagorean theorem relates all three: a² + b² = c². Given any two of the three sides, you can always solve for the third.

How the calculation works

Enter two of the three side lengths and leave the unknown one blank. If the hypotenuse (c) is unknown, the calculator adds the squares of the legs and takes the square root: c = √(a² + b²). If a leg is unknown, it rearranges the theorem: a = √(c² − b²) or b = √(c² − a²). Once all three sides are known, the calculator also reports the triangle's area (½ × a × b, since the legs are perpendicular), its perimeter (a + b + c), and its two non-right interior angles, found with inverse sine: angle opposite a = sin⁻¹(a ⁄ c) and angle opposite b = 90° minus that value.

Common mistakes

  • Squaring instead of adding first: a² + b² = c² means square each leg separately, then add, then take the square root of the sum — not (a + b)².
  • Mixing up the hypotenuse: the hypotenuse is always the side opposite the right angle and always the longest side. If your "unknown" side isn't actually the longest, you may be solving for a leg, not the hypotenuse.
  • Impossible triangles: a leg can never be longer than the hypotenuse. If you enter a leg and a hypotenuse where the leg is greater than or equal to the hypotenuse, no valid right triangle exists (c² − b² would be zero or negative).

Real-world applications

  • Construction and carpentry use the 3-4-5 rule (or any multiple, like 6-8-10) to square up corners, foundations, and framing without a protractor.
  • Ladder and ramp safety calculations use the theorem to find safe height or horizontal reach from a fixed length.
  • Navigation and surveying use it to find straight-line (as-the-crow-flies) distance from two perpendicular legs, such as north-south and east-west displacement.
  • Screen and TV sizing uses the diagonal as the hypotenuse of the width and height of the display.

Frequently Asked Questions

What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse: a² + b² = c², where c is the side opposite the right angle. For example, legs of 3 and 4 give a hypotenuse of √(3² + 4²) = 5.
How do I find a missing leg instead of the hypotenuse?
Rearrange the theorem to solve for a leg: a = √(c² − b²) or b = √(c² − a²). Leave the leg you want to solve for blank and provide the hypotenuse and the other leg.
How is the area of a right triangle calculated?
Since the two legs of a right triangle are perpendicular, one can be treated as the base and the other as the height: Area = ½ × a × b. A 3-4-5 triangle has an area of ½ × 3 × 4 = 6 square units.
Can this calculator check whether a triangle is a right triangle?
Yes. Enter all three side lengths without leaving any field blank. The calculator checks whether a² + b² equals c² (within rounding tolerance) and tells you whether the triangle is a valid right triangle.

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