What Is the Pythagorean Theorem Used For?

The Pythagorean Theorem is used to find the length of the hypotenuse of a right triangle, a calculation which affords many practical uses, such as within the fields of construction, land surveying and navigation. The relationship between the two legs of a right triangle and the hypotenuse, shown by the equation a2 + b2 = c2, is known as the Pythagorean triplet, and its use in ancient megalithic construction is believed to predate the discovery of writing. The ancient Egyptians used a rope marked in the Pythagorean triples of 3, 4 and 5 to create right triangles and some evidence points to a possible use by Babylonian mathematicians.

Modern builders may mark adjacent lengths of a frame 6 feet and 8 feet from a corner, and then adjust the length between the end points to 10 feet to meet the relationship of the Pythagorean triplet 6-8-10. When all three of the values of a Pythagorean triplet are divisible solely by the greatest common denominator of 1, the triplet is said to be coprime or primitive. Some examples of primitive Pythagorean triplets are 3-4-5, 5-12-13 and 7-24-25.

Another way of expressing the Pythagorean Theorem is that the hypotenuse of a right triangle is equal to the square root of the sum of the squares of its legs. A useful application of this formula is derived from rearranging it so that the length of any one of the three sides of a right triangle can be determined, provided that the lengths of the other two sides are known.