Visualization of the Pythagorean Theorem

 

Many remember the Pythagorean Theorem that they learned in middle school. It goes A^2 + B^2 = C^2. It is a theorem about right triangles, triangles with one angle equal to 90 degrees. There are 3 sides to a right triangle: the 2 legs, adjacent to the right angle, and 1 hypotenuse, the side opposite the right angle and also the longest side of a right triangle. In the equation above, A and B are the lengths of the legs of the right triangle, and C is the length of the hypotenuse. So, what A^2 + B^2 = C^2 represents is A multiplied by itself plus B multiplied by itself is equivalent to C multiplied by itself. This can be represented visually as the area of squares because the area of a square is the length of one side length multiplied by itself.

 

In the figure below, there are two equally sized big squares, each containing 4 congruent (the same) yellow triangles. In the figure on the left, there is a red square in the middle. The side length of the square is C, the hypotenuse, and so the area of the square is C^2 or the length of the hypotenuse multiplied by itself.

 

In the figure on the right, there is a blue square and a green square. The blue square's area is B^2 because B is its side length and also the length of one of the longer leg of the right yellow triangle. The green square's area is A^2 because A is its side length and is also the length of the shorter leg of the right yellow triangles.

 

To say that A^2 + B^2 = C^2, after realizing that both big squares are equal in area, we can remove the 4 right triangles from both sides and to find that we are left with equivalent areas in both figures. On one side is the red square, C^2, and on the other side are the blue and the green squares, A^2 and B^2. The area of this side is then A^2 + B^2. And because the areas of both sides are equivalent C^2 = A^2 + B^2.