Pythagorean Theorem

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
c2 = a2 + b2

Try this Drag the orange dots on each vertex of the right triangle below. The formula showing the calculation of the Pythagorean Theorem will change accordingly.

Although Pythagoras' name is attached to this theorem, it was actually known centuries before his time by the Babylonians. There are many proofs of this theorem, some graphical in nature and others using algebra. See A graphical proof of the Pythagorean Theorem for one such proof.

On the web site "cut-the-knot", the author collects proofs of the Pythagorean Theorem, and as of this writing has listed over 70, but hundreds are actually known.

Solving the right triangle

The term "solving the triangle" means that if we start with a right triangle and know any two sides, we can find, or 'solve for', the unknown side. This involves a simple re-arrangement of the Pythagoras Theorem formula to put the unknown on the left side of the equation.

Find the hypotenuse

If we know the two legs of a right triangle we can solve for the hypotenuse using the formula: where a and b are the lengths of the two legs of the triangle, and
h is the hypotenuse.

Find a leg

If we know the hypotenuse and one leg, we can find the other leg using the formula: where a is the leg we wish to find
b is the known leg
h is the hypotenuse.

The Converse of the Pythagorean Theorem

The converse of this theorem is also true. That is, if a triangle satisfies Pythagoras' theorem, then it is a right triangle. Put another way, only right triangles will satisfy the theorem.

Things to try

  1. In the figure above, click on 'reset'.
  2. Check one of the 'hide' checkboxes.
  3. Adjust the triangle by dragging an orange dot.
  4. Use the Pythagorean Theorem to find the missing side.
  5. Uncheck the 'hide' box to check your answer.

Other triangle topics


Perimeter / Area

Triangle types

Triangle centers

Congruence and Similarity

Solving triangles

Triangle quizzes and exercises