Understanding the Pythagorean Theorem

Let’s pretend you are building a wheelchair ramp into your home and you need to know how many feet long your ramp is going to be so that you may buy the right amount of materials. How will you figure that out?

The best way is by utilizing the Pythagorean theorem. A theorem is a mathematical equation which has been proved to work time after time. This one in particular pertains to right triangles and the measurements of their sides. What is a right triangle? Think of a triangle which has one angle or corner which fits perfectly into the corner of a looseleaf sheet of paper.

Pythagoras (572-407 BC), for whom this theorem is named, was the leader of a society or school which combined religion and philosophy to explore the areas of science, music, mathematics, and astronomy. They sought order in the chaotic world they saw around them. Many of the ideas, including possibly this one, were attributed to Pythagoras because he was the leader. The idea upon which the theorem is based was known by the Sumerians of ancient times, but Pythagoras and his society were the ones to prove the equation. In a book no longer in print called “The Pythagorean Proposition,” author Elisha Scott Loomis demonstrates 256 proofs to show this equation to be always true.

The Pythagorean theory gives us a foolproof way to figure out the length of one side of a right triangle if the measures of the lengths of two of the sides are already known. The mathematical equation is “a squared plus b squared equals c squared.” The variables a and b stand for the measures of the two shorter legs of the right triangle and c represents the longest side, known as the hypotenuse. When this theorem is applied to right triangles we find certain numbers to always go together. These are known as Pythagorean triples.

So what, you may say. When will I need this information? Let’s go beyond the normal textbook problems of distances as the crow flies and heights of mountains to more practical applications.

Take our original problem. When built, the ramp surface will be the hypotenuse of a right triangle. The ground surface from the end of the ramp to the door is one of the shorter legs of the triangle that will be formed. From the foundation of the building to the door is the other short leg. Both short legs of the right triangle are measurable. Let’s say that the foundation to the door measures 3 feet. You want the ramp to begin 4 feet from the door. Plugging those numbers into the equation gives you (3) squared plus (4) squared equals c squared. (3) squared equals 9; (4) squared equals 16 ; 9 plus 16 equals 25. What number multiplied by itself equals 25? The answer is 5, so the ramp surface will be 5 feet long.

This theorem may also be utilized to lay out stakes for a rectangular garden plot or the foundation for a building. First mark where corner A is going to be. Place a second marker (B) 6 yards from marker A. To make a square corner, plan the third mark (C) to be 8 yards from A and 10 yards from B. Why? Because (6) squared equals 36, (8) squared equals 64, 36 plus 64 equals 100, and the square root of 100 is 10. To finish the rectangle, make mark D 8 yards from B and 6 yards from C.

You may, of course, use your own measurements for the legs of the right triangle and as long as you know the length of two of the legs, you will always be able to figure out the third.