Write down all the known elements of your right triangle. For instance, you may know the value only of the hypotenuse and one other side. You may know "a" and "b" but not the hypotenuse. Alternatively, you may know the value for only one side of the triangle or none at all. Sketch a new right triangle and fill in, as much as you can, the values for "a," "b" and "c."
There may be times when you can deduce the value of a particular side. For instance, if you know the value of "c" but do not know the values of "a" and "b," look for the presence of adjacent squares. If the area of an adjacent square is known, you can find its square root to determine the length of "a" or "b." For example, if the area of the square adjacent to "a" was 25, then its square root --- and, by extension, "a" --- would equal 5. Likewise, if you know that your right triangle has two angles that are 45 degrees, you know that the "a" and "b" sides have equivalent lengths.
Once you have determined the length of at least two sides, you can replace the values in the equation to solve the Pythagorean theorem. For instance, if "c" was 10 in the previous example, you could write: 5^2 + b^2 = 10^2. If you square the known values, you get: 25 + b^2 = 100. To solve for any variable, isolate it on one side of the equation, in this case, by subtracting 25 from both sides to get: b^2 = 100 - 25, or b^2 = 75.
In the previous example, the value of "b" is known only in relation to its square, or "b times b," so you must find the square root to know that "b = 8.661." In some cases, you may know the square root of only two variables. For instance, if the square root of both "a" and "b" were 2, you could write: 2^2 + 2^2 = c^2. In this case, you would have to find the squares of "a" and "b" before you could solve for "c": 4 + 4 = c^2, or 8 = c^2. In this case, the square root of "c" would be 2.823.