Pythagorean Theorem: Formula, Triples and Uses

The Pythagorean theorem says that in a right triangle the square of the longest side (the hypotenuse, c) equals the sum of the squares of the two shorter sides (the legs, a and b): a squared + b squared = c squared. It holds only when one angle is exactly 90 degrees, and it lets you find any one side when you know the other two.

How do you solve for each side?

• Hypotenuse: c = square root of (a squared + b squared)
• A missing leg: a = square root of (c squared - b squared), or b = square root of (c squared - a squared)
• Check: the hypotenuse is always the longest side, so c must be larger than either leg

Example: legs of 5 and 12 give c = square root of (25 + 144) = square root of 169 = 13. The area is half the product of the legs, 0.5 x 5 x 12 = 30, and the perimeter is 5 + 12 + 13 = 30. The two acute angles come from the tangent: atan(5 / 12) = 22.62 degrees opposite the 5 side and 90 - 22.62 = 67.38 degrees opposite the 12 side. These are the same quantities the Right Triangle Calculator reports.

How can you square a corner with it?

If you mark out a rectangle 12 feet by 16 feet, both diagonals must measure exactly square root of (144 + 256) = 20 feet for the corners to be true right angles. A diagonal that is off tells you how far out of square the corner is, using the law of cosines (the general form of the theorem for any triangle).

Half a foot of diagonal error across a 12 by 16 layout is therefore about 3 degrees, which is plenty to see in a deck or frame. Measurement error matters as much as arithmetic: measure to the same point on the tape each time. For structural work, follow your local building code and the plans rather than a blog post.

Where else does the theorem turn up?

The screen row works backwards: for a 16:9 shape the diagonal is square root of (16 squared + 9 squared) = 18.358 units per 16 units of width, so width = 27 x 16 / 18.358 = 23.53 inches. Three-dimensional diagonals use the theorem twice, which is why the box diagonal is the square root of the sum of three squares. For a ladder, the right base distance depends on the manufacturer's instructions and safety guidance, so use this figure for geometry only.

What are Pythagorean triples?

A Pythagorean triple is a set of three whole numbers that satisfy the equation exactly, such as 3-4-5 and 5-12-13. Multiplying every number in a triple by the same factor gives another (3-4-5 becomes 6-8-10). For any whole numbers m greater than n, the three numbers m squared - n squared, 2mn and m squared + n squared always form a triple, because (m squared - n squared) squared + (2mn) squared simplifies to (m squared + n squared) squared. This is Euclid's formula.

Does it work in reverse?

Yes. If three sides satisfy a squared + b squared = c squared, the triangle is a right triangle. If c squared is smaller than a squared + b squared, the largest angle is acute; if it is larger, that angle is obtuse. Sides 6, 8, 10 give 36 + 64 = 100, a right triangle. Sides 7, 9, 11 give 49 + 81 = 130, which is more than 121, so all angles are acute. Sides 6, 8, 11 give 100, which is less than 121, so one angle is obtuse.

Common mistakes

• Using the theorem on a triangle that is not a right triangle. For other triangles use the law of cosines: c squared = a squared + b squared - 2ab cos(C). When C is 90 degrees, cos(C) is 0 and the extra term vanishes.
• Adding the lengths instead of the squares, or forgetting the final square root.
• Treating a leg as the hypotenuse. Solving for a leg needs a subtraction, and the result is only real if the hypotenuse is the larger value; the calculator returns nothing when c is not larger than the known leg.
• Mixing units. Feet and inches must be converted to one unit before squaring.
• Over-rounding early. Carry several decimals through the square root and round at the end.