The Pythagorean Theorem is named after the ancient Greek mathematician, Pythagoras (~500 B.C.). This theorem is used to find the third side of any right triangle when two sides are known. The Pythagorean Theorem states that a^{2} + b^{2} = c^{2}. [Figure 1-32] Where c = the hypotenuse of a right triangle, a is one side of the triangle and b is the other side of the triangle.

Example: What is the length of the longest side of a right triangle, given the other sides are 7 inches and 9 inches? The longest side of a right triangle is always side c, the hypotenuse. Use the Pythagorean Theorem to solve for the length of side c as follows:

If c^{2} = 130 then c = √130 = 11.4 inches, therefore, side c = 11.4 inches.

Example: The cargo door opening in a military airplane is a rectangle that is 5 1⁄2 feet tall by 7 feet wide. A section of square steel plate that is 8 feet wide by 8 feet tall by 1 inch thick must fit inside the airplane. Can the square section of steel plate fit through the cargo door? It is obvious that the square steel plate will not fit horizontally through the cargo door. The steel plate is 8 feet wide and the cargo door is only 7 feet wide. However, if the steel plate is tilted diagonally, will it fit through the cargo door opening?

The diagonal distance across the cargo door opening can be calculated using the Pythagorean Theorem where “a” is the cargo door height, “b” is the cargo door width, and “c” is the diagonal distance across the cargo door opening.

The diagonal distance across the cargo door opening is 8.9 feet, so the 8-foot wide square steel plate will fit diagonally through the cargo door opening and into the airplane.