A graphical proof of the Pythagorean Theorem

This graphical 'proof' of the Pythagorean Theorem starts with the right triangle below, which has sides of length a, b and c. It demonstrates that  a2 + b2 = c2,  which is the Pythagorean Theorem.

It is not strictly a proof, since it does not prove every step (for example it does not prove that the empty squares really are squares). But it does demonstrate the theorem in an interesting way.

Instructions Click on 'Next' to go through the proof one step at a time, or click on 'Run' to let it run without stopping.

The proof step-by-step

Step 1 Make 3 copies of the original triangle and arrange the four triangles in a square as shown. The outer square JKLM will remain fixed throughout the rest of the proof.
Step 2 Each side of the empty square in the middle has a length of c, and so has an area of c2.
Step 3 Re-arrange the triangles as shown so that the empty space is now divided into two smaller squares.
Step 4 Notice that the top left empty square has each side equal to a, so its area is a2.
Step 5 Notice also that the bottom right empty square has each side equal to b,
so its area is b2.
Step 6 Done. We have rearranged the triangles inside a constant-size square. The empty space we started with ( c2 ) must be equal to the sum of the two empty spaces at the end.

Therefore  a2+b2 = c2    QED.

Other triangle topics


Perimeter / Area

Triangle types

Triangle centers

Congruence and Similarity

Solving triangles

Triangle quizzes and exercises