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The Pythagorean Theorem describes the lengths of the sides of a right triangle in a way that is so elegant and practical that the theorem is still widely used today. The theorem states that for any right triangle, the sum of the squares of the non-hypotenuse sides is equal to the square of the hypotenuse. In other words, for a right triangle with perpendicular sides of length a and b and hypotenuse of length c, a2 + b2 = c2. The Pythagorean Theorem is one of the fundamental pillars of basic geometry, having countless practical applications - using the theorem, for instance, it's easy to find the distance between two points on a coordinate plane.

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1. Ensure that your triangle is a right triangle. The Pythagorean Theorem is applicable only to right triangles, so, before proceeding, it's important to make sure your triangle fits the definition of a right triangle. Luckily, there is only one qualifying factor - to be a right triangle, your triangle must contain one angle of exactly 90 degrees.[1]
As a form of visual shorthand, right angles are often marked with a small square, rather than a rounded "curve", to identify them as such. Look for this special mark in one of the corners of your triangle.
2. Assign the variables a, b, and c to the sides of your triangle. In the Pythagorean Theorem, the variables a and b refer to the sides that meet in a right angle, while the variable c refers to the hypotenuse - the longest side which is always opposite the right angle. So, to begin, assign the shorter sides of your triangle the variables a and b (it doesn't matter which side is labeled 'a' or 'b'), and assign the hypotenuse the variable c.[2]
3. Determine which side(s) of the triangle you are solving for. The Pythagorean Theorem allows mathematicians to find the length of any one of a right triangle's sides as long as they know the lengths of the other two sides. Determine which of your sides has an unknown length - a, b, and/or c. If the length of only one of your sides is unknown, you're ready to proceed.[3]
Let's say, for example, that we know that our hypotenuse has a length of 5 and one of the other sides has a length of 3, but we're not sure what the length of the third side is. In this case, we know we're solving for the length of the third side, and, because we know the lengths of the other two, we're ready to go! We'll return to this example problem in the following steps.
If the lengths of two of your sides are unknown, you'll need to determine the length of one more side to use the Pythagorean Theorem. Basic trigonometry functions can help you here if you know one of the non-right angles in the triangle.
4. Plug your two known values into the equation. Insert your values for the lengths of the sides of your triangle into the equation a2 + b2 = c2. Remember that a and b are the non hypotenuse sides, while c is the hypotenuse.[4]
In our example, we know the length of one side and the hypotenuse (3 & 5), so we would write our equation as 3² + b² = 5²
5. Calculate the squares. To solve your equation, begin by taking the square of each of your known sides. Alternatively, if you find it easier, you may leave your side lengths in the exponent form, then square them later.[5]
In our example, we would square 3 and 5 to get 9 and 25, respectively. We can rewrite our equation as 9 + b² = 25.