by Kerry Kauffman

Created on: June 20, 2008   Last Updated: July 08, 2008

The Pythagorean theorem is a theorem encountered in geometry and is a very useful way to determine the lengths of the sides of a right triangle. A right triangle is a triangle with a 90 degree or "right" angle. The other two angles will add up to 90 degrees since all triangles have their angle measures adding up to 180.

Think of the right triangle having 3 sides, 2 of which called "legs" form the right angle and the side opposite the right angle, called the hypotenuse. In words the Pythagorean theorem is length of a leg squared plus length of the other leg squared equals the hypotenuse squared. As an algebraic equation the Pythagorean theorems reads a^2 + b^2 = c^2 (a- squared plus b- squared equals c- squared).

For example, suppose a right triangle has sides a and b equalling 4 and 6 respectively. Using the Pythagorean theorem you can get the length of side c, the hypotenuse. First square 4 to get 16, square 6 to get 36 and add them together to get 52. Taking the square root of 52 gives the length of the hypotenuse as 7.2.

Some combinations of the lenghts of the sides and hypotenuse form what is called a Pythagorean triple. This is when the lengths of the sides are all whole numbers. One example is a right triangle with sides 3 and 4, using the Pythagoren theorem yields 3^2+4^2=c^2, which makes c=5. So (3,4,5) is a Pythagorean triple. Likewise and multiple of 3,4,5 such as (6,8,10) and (9,12,15) etc. More Pythagorean triples are (5,12,13), (7,24,25) and any multiples of both. Knowing the Pythagorean triples saves one from having to use the Phythagorean theorem to calculate the length of the missing side. Note that any 1 of the 3 sides can be missing and the Pythagorean theorem can be used.

One important thing to note, the Pythagorean theorem can ONLY be used with right triangles. For any other type of triangle, Trigonometric methods need to be used to calculate the length of the sides. Those methods are sine, cosine and tangent and also involve the measure of the angles of the triangle.

A practical use of the Pythagorean theorem would be in the case of removing stair and putting in a ramp. Let's say that the height of the staircase is 6 feet and the length of the staircase is 9 feet. Removing the staircase and putting in a ramp will require the theorem to figure out the length of the ramp needed. In this case the length would be the square root of 117, which is 10.8 feet.

All students who take Geometry will learn the Pythogorean theorem and will be used throughout Algebra, Trigonometry and Physics to name a few. It's definitely something worthwhile learning and understanding and I hope my explanation and examples make it easier to understand.

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