Take a look at this right triangle. A right triangle is simply a triangle that contains a right angle (90°). The longest side is called the hypotenuse, and we often denote it "c." The other two sides are called legs, and we often denote them "a" and "b."
Assuming that we have labeled our triangle in that way, the following theorem applies. In words, the square of leg "a" plus the square of leg "b" equals the square of hypotenuse "c." That is all there is to it.
Typically in a right triangle problem we are given the lengths of two out the three sides, and we must find the missing side. It can be any of the three, so we have to remember substitute in the formula correctly.
Here is an example. Assume we have a triangle with legs of length 3 and 4. We need to find the hypotenuse. Sometimes you will get this information in a word problem like this, and sometimes you will just be given a diagram. In this case our missing side is "c." Look at the picture at left to see how the formula is being used. The first step is substitution, in this case, the known values of "a" and "b." The next step is evaluating the squares. Remember, to square a number means to multiply it by itself. Then add the squares.
Many students gets confused with the next step. We don't yet know the value of "c." We just know that c² equals 25. Hold that thought and see the next step.
In math, if we take the square root of any square, we get back to the original number. This is because squaring and "square rooting" are inverse operations (opposites). They "undo" each other. There is a bit of "fine print" along with that statement, but in this context it applies just fine.
With that stated, since we want the value of "c" and not c², we must take the square root of c-sq. Whatever we do to one side of an algebraic equation, we must do to the other, as shown. We are left with c = 5, which is our answer. Note that if we were familiar with something known as Pythagorean Triples, we could have actually gotten this answer without doing any work, although sometimes we are specifically told to show our work. See the Resource section for more about Triples.
In problems like this, always check your answer for reasonableness. Picture a right triangle with one leg measuring 3 units and one measuring 4 units. A hypotenuse of 5 units seems reasonable. If you got an answer of 1.7 or 938, surely you made some mistake.
Here is another example that we'll step through more quickly. A right triangle has legs of 7 and 5. Find the hypotenuse. See the equation steps at left. In this case, we end up with c = sqrt(74). Very often we won't end up with an integer (whole number) answer to our problem. It turns out that sqrt(74) cannot be simplified so we must leave it as is. See the Resource section for more about simplifying square roots. Sometimes we are told to just evaluate the square root on a calculator, and round our answer to a given place value. In this case it was rounded to the nearest tenth.
One more example. A right triangle has a leg of 5, and a hypotenuse of 13. Find the missing leg. Be careful! In this case, we are given "a" and "c," and "b" is missing. See the steps at left. We'll still use the formula the same way, but to get from line 2 to line 3, we have to subtract 25 from each side, which most algebra students feel comfortable with. The rest is as before.
That's basically all there is to the Pythagorean Theorem. Obviously memorize it and practice using it since it such a "hot" topic among teachers and test makers, and it's actually quite straightforward.