Pythagorean Theorem Calculator

Find the missing side of a right triangle from the other two — the hypotenuse or either leg — with every step worked, plus the area, perimeter and angles.

Your details

Any unit, the same for every side.

Any unit, the same for every side.

Hypotenuse c

5

Area
6
Perimeter
12
Angle A (opposite a)
36.87 °
Angle B (opposite b)
53.13 °
  • The theorem holds for right triangles only. The sides have no unit here: use the same unit for every side and the answers come out in it.

Visual breakdown

  • Side a
  • Side b
  • Hypotenuse c

  1. 1

    Square the two legs

    a² and b²

    = 3² = 9, 4² = 16

    = 9 and 16

  2. 2

    Add them

    c² = a² + b²

    = 9 + 16

    = 25

  3. 3

    Take the square root

    c = √(a² + b²)

    = √25

    = 5

  4. 4

    Area

    A = ½ × a × b

    = ½ × 3 × 4

    = 6

  5. 5

    Perimeter

    a + b + c

    = 3 + 4 + 5

    = 12

  6. 6

    The angles

    A = arctan(a ÷ b), B = 90° − A

    = arctan(3 ÷ 4)

    = 36.87° and 53.13°

A right triangle with legs of 3 and 4 has a hypotenuse of 5. The area is 6, the perimeter 12, and the angles are 36.87° and 53.13°.

What this calculates

Choose which side you want, enter the other two, and this works out the missing side of a right triangle with every step shown, then the area, the perimeter and the two acute angles. It uses a² + b² = c², where c is the hypotenuse. The sides can be in any unit, as long as it is the same for all of them.

The formula

Hypotenuse

c = √(a² + b²)

A leg

a = √(c² − b²), b = √(c² − a²)

Angles

A = arctan(a ÷ b), B = 90° − A

The hypotenuse is always the longest side, so when you solve for a leg it must be shorter than the hypotenuse.

Worked example

Finding a leg: with a hypotenuse of 13 and a leg of 12, the other leg is √(169 − 144) = 5.

When to use it, and the mistakes to avoid

Use it for the length of a diagonal, a ladder against a wall, a roof's rafter, the distance between two points on a grid, or to check that a corner is square.

The mistakes that cost the most:

  • Using it on a triangle with no right angle. It only holds for 90°.
  • Adding the sides instead of their squares. c is not a + b.
  • Forgetting the square root. √25 is 5; 25 is c².
  • Solving a leg with a hypotenuse that is too short. The hypotenuse is the longest side.
  • Mixing units. Convert every side to one unit first.

FAQ

What is the Pythagorean theorem?

In a right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². With legs of 3 and 4, c² = 9 + 16 = 25, so the hypotenuse is 5.

How do I find a missing leg?

Rearrange the formula: a = √(c² − b²). With a hypotenuse of 13 and one leg of 12, a = √(169 − 144) = √25 = 5. The hypotenuse must be longer than the leg you know, or no triangle exists.

Does it work for any triangle?

No, only for right triangles, where one angle is exactly 90°. For other triangles you need the law of cosines. A handy test: if a² + b² = c² holds for three sides, the triangle has a right angle.

What is a Pythagorean triple?

Three whole numbers that fit a² + b² = c², such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Any multiple of a triple is another, so 6-8-10 and 9-12-15 work too. Builders use 3-4-5 to check a corner is square.

What unit are the answers in?

The same as the sides you enter. The theorem does not depend on the unit, so centimetres in give centimetres out and feet in give feet out. Use one unit for every side. The area comes out in that unit squared.

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