Pythagorean Theorem: How to Use It and When It Works

A right triangle contains one angle of 90 degrees. The side opposite that right angle is the hypotenuse, and it is always the longest side. In the formula a squared plus b squared equals c squared, c must represent the hypotenuse. The letters a and b can be assigned to the other two sides in either order.

Suppose the shorter sides are 3 and 4. Squaring them gives 9 and 16, and their sum is 25. The square root of 25 is 5, so the hypotenuse is 5. This familiar 3-4-5 triangle is a quick example of the Pythagorean theorem in action.

The theorem is not a general rule for every triangle. If there is no right angle, the relationship does not hold in this simple form. Checking the angle information before calculating prevents one of the most common mistakes.

If the missing side is the hypotenuse, add the squares of the two shorter sides and then take the square root. For a triangle with legs 6 and 8, you calculate 36 + 64 = 100, so the hypotenuse is 10.

If the hypotenuse is known and a shorter side is missing, rearrange by subtraction. If c = 13 and one shorter side is 5, then the missing side squared is 13 squared minus 5 squared, or 169 - 25 = 144. The missing length is 12.

This is why the Pythagorean theorem is easier when you label the diagram before touching the calculator. Find the right angle, mark the hypotenuse, decide which side is missing, and then choose addition or subtraction. The structure of the triangle tells you the operation.

Before you finish a problem, ask:

• •Is the triangle definitely a right triangle?
• •Have I identified the hypotenuse as the side opposite the right angle?
• •Did I square each known side before adding or subtracting?
• •Did I take a square root at the end?
• •Does the answer make sense, especially compared with the hypotenuse?

The theorem is also useful in coordinate geometry. If two points differ by 3 units horizontally and 4 units vertically, those changes form the shorter sides of a right triangle. The straight-line distance between the points is the hypotenuse, so the Pythagorean theorem gives a distance of 5 units. The same idea appears in navigation, construction, and diagonal measurements. Whenever perpendicular directions create a right triangle, the theorem can connect the two component distances with the direct distance between their endpoints.

You can also use the relationship in reverse. If three side lengths satisfy the equation with the largest side as c, the triangle is a right triangle. This makes the theorem useful for checking right angles as well as finding missing lengths.

One of the simplest proofs uses rearrangement. Draw a large square with sides of length a plus b. Inside it, place four copies of your right triangle, one in each corner, so that their hypotenuses form a tilted square in the middle with side c. The area of the big square is (a + b) squared, which expands to a squared plus 2ab plus b squared. It also equals four triangles, each with area ab divided by 2, plus the central square c squared, giving 2ab plus c squared. Remove 2ab from both expressions and a squared plus b squared equals c squared.

The relationship is much older than Pythagoras. A Babylonian clay tablet known as Plimpton 322, dating from roughly 1800 BCE, contains columns of numbers usually interpreted as relating to Pythagorean triples, and the relationship also appears in early Chinese mathematical texts. No writings by Pythagoras himself survive, and the story that he proved the theorem comes from authors who lived centuries after him. His name became attached to the result through Greek tradition, which valued general proof, but people in several cultures were using the relationship long before.

Television and phone screens are sold by the length of their diagonal, so the theorem tells you how big a screen really is. A widescreen display has sides in the ratio 16 to 9. Treat those as the shorter sides of a right triangle: 16 squared plus 9 squared is 256 plus 81, which is 337, so the diagonal is the square root of 337, about 18.36 units. A 55 inch screen therefore has a width of 55 times 16 divided by 18.36, about 47.9 inches, and a height of 55 times 9 divided by 18.36, about 27 inches.

The theorem also works in three dimensions if you use it twice. Will a pole 120 centimetres long fit inside a box measuring 80 by 60 by 50 centimetres? First find the diagonal across the base: 80 squared plus 60 squared is 6,400 plus 3,600, which is 10,000, so the base diagonal is 100 centimetres. That diagonal and the height of the box form another right triangle, so the longest straight line inside the box is the square root of 10,000 plus 2,500, which is about 111.8 centimetres. The pole will not fit, even placed corner to corner.

Whole-number solutions such as 3, 4, 5 are called Pythagorean triples, and a recipe set out in Euclid's Elements produces as many as you like. Choose two whole numbers m and n, with m larger than n. The three sides are m squared minus n squared, 2 times m times n, and m squared plus n squared. With m equal to 2 and n equal to 1, you get 3, 4 and 5. With 3 and 2 you get 5, 12 and 13, and with 4 and 1 you get 15, 8 and 17. Check the last one: 225 plus 64 is 289, which is 17 squared.

Mathematicians naturally asked what happens with higher powers. Could two whole-number cubes add up to another cube, or two fourth powers to a fourth power? In the seventeenth century Pierre de Fermat wrote in the margin of a book that no such solutions exist for any power above 2, adding that he had a marvellous proof which the margin was too small to contain. The claim, known as Fermat's Last Theorem, resisted every attempt for more than 350 years until Andrew Wiles completed a proof in the mid-1990s, using mathematics far beyond anything available to Fermat.

The Pythagorean theorem connects the three side lengths of a right triangle through a squared plus b squared equals c squared. Label the hypotenuse correctly, decide whether you are finding the hypotenuse or a shorter side, and keep the squared quantities separate until the final square root. A good diagram usually makes the calculation much easier.