The Pythagorean Theorem

What the theorem says

A right triangle is a triangle with one 90° angle, a “square corner.” Its two shorter sides meet at that corner. They are called the legs, and we usually name them a and b.

The side across from the right angle is the hypotenuse, named c. It is always the longest side.

a² + b² = c²

“Square” has two meanings here, and both work. In numbers, 3² means 3 × 3 = 9. In pictures, 3² is the area of a square whose sides are 3 long. So the theorem also says: build a square on each side of a right triangle, and the two small squares together have exactly the same area as the big one.

One warning: this rule is only for right triangles. For other triangles, a² + b² and c² will not match.

Why should this be true for every right triangle? Here is a classic rearrangement proof. You need four copies of the same right triangle and a square frame whose sides are a + b long.

1. Both big squares are the same size: a + b on each side.
2. Both hold the same four triangles, so the triangles cover the same area each time.
3. Take the triangles away and what is left must be equal too. On the left that’s c². On the right it’s a² + b².
4. So a² + b² = c². Nothing in the argument depended on the actual lengths, so it works for every right triangle.

One detail worth checking: the middle shape on the left really is a square. Its four sides are all c. At each corner, the two acute angles of a triangle add to 90°, so the angle left over on the straight edge is 180° − 90° = 90°.

Worked examples

Example 1: Find the hypotenuse

A TV screen is 48 inches wide and 27 inches tall. TVs are sold by the length of the diagonal. What size is it?

1. The width and height meet at a right angle, so they are the legs: a = 48, b = 27.
2. c² = 48² + 27² = 2304 + 729 = 3033
3. c = √3033 ≈ 55.1 inches. It would be sold as a 55-inch TV.

Example 2: Find a missing leg

A 13-foot ladder leans against a wall. Its foot is 5 feet from the wall. How high up the wall does it reach?

1. The ladder is across from the right angle (where wall meets ground), so it is the hypotenuse: c = 13. One leg is a = 5.
2. Rearrange: b² = c² − a² = 169 − 25 = 144
3. b = √144 = 12 feet.

Example 3: When the answer isn’t whole

Legs of 5 and 7: c² = 25 + 49 = 74, so c = √74 ≈ 8.6. Most right triangles look like this. Leave the answer as √74 for an exact value, or round it when you need a measurement.

The converse: is this triangle right?

The theorem also works backwards. This is called its converse: if the sides of a triangle fit a² + b² = c², then the triangle must have a right angle, opposite side c.

To test a triangle, always use the longest side as c.

The table shows one more thing. If a² + b² comes out bigger than c², the largest angle is acute; if it comes out smaller, that angle is obtuse.

Euclid proves this converse right after the theorem, as Book I, Proposition 48.

Pythagorean triples

A Pythagorean triple is three whole numbers that fit a² + b² = c². They are worth recognizing on sight, because they make problems come out clean.

Multiply every number in a triple by the same whole number and you get another one: 3-4-5 becomes 6-8-10, 9-12-15, 30-40-50. Triples with no common factor, like 3-4-5, are called primitive.

Want to make your own? Pick whole numbers m > n > 0. Then m² − n², 2mn, and m² + n² always form a triple. With m = 2, n = 1 you get 3, 4, 5. Try m = 3, n = 2.

Where you’ll actually use it

• Building square corners. Builders mark 3 feet along one wall and 4 feet along the other. If the marks are exactly 5 feet apart, the corner is a true right angle. (That’s the converse at work.)
• Screens. Phone, laptop and TV sizes are diagonal measurements, found from width and height, as in Example 1.
• Distance on a map or grid. Walk 3 blocks east and 4 blocks north and you are 5 blocks from the start “as the crow flies.” The distance formula you’ll meet in coordinate geometry is this theorem in disguise.
• Ramps, ladders and roofs. Any time a slope connects a horizontal run to a vertical rise, the slope’s length is a hypotenuse.
• 3D. Use it twice to find the diagonal of a box: d² = l² + w² + h².

Common mistakes

A short history: older than Pythagoras

The theorem carries the name of Pythagoras, a Greek thinker of the 6th century BCE. But people were using the rule more than a thousand years before him.

The best-known evidence is a clay tablet called Plimpton 322, written in cuneiform in the Old Babylonian period, around 1800 BCE, probably in southern Iraq. It is now in the collection of Columbia University in New York. Its 15 rows of numbers are tied to what we now call Pythagorean triples, some of them large: one row corresponds to the triangle with sides 119, 120 and 169. Scholars still debate exactly why it was made; one leading view is that it helped a teacher set problems about right triangles.

Indian and Chinese mathematicians also stated the rule. The Baudhāyana Śulba Sūtra (roughly 800–500 BCE) gives it for the diagonal of a rectangle. The Chinese Zhoubi Suanjing, compiled between about 100 BCE and 100 CE, states it as the “gougu” rule, and a 3rd-century CE commentary on it by Zhao Shuang adds a diagram built on a 3-4-5 triangle (see Zhoubi Suanjing on Wikipedia).

So why “Pythagoras”? Nothing he wrote survives, and we know about him mainly from writers who lived centuries later. Linking the theorem to him is a tradition, not a documented fact. Some historians think he or his followers may have been the first to prove it in general, rather than just use it. The oldest proof that survives in Greek is in Euclid’s Elements (around 300 BCE), Book I, Proposition 47.

You’ll get three kinds: find the hypotenuse, find a leg, and decide whether a triangle is right. When the answer isn’t a whole number, round to the nearest tenth or type the exact root, like √74 or sqrt(74).

Sources

• Euclid, Elements, Book I, Propositions 47 and 48, ed. D. E. Joyce, Clark University: Euclid’s proof of the Pythagorean theorem (Elements I.47) and its converse (Elements I.48).
• J. J. O’Connor and E. F. Robertson, MacTutor History of Mathematics, University of St Andrews: Pythagoras of Samos (biography) and Pythagoras’s theorem in Babylonian mathematics.
• Eleanor Robson, “Words and Pictures: New Light on Plimpton 322,” American Mathematical Monthly 109, no. 2 (2002): 105–120, doi:10.2307/2695324.
• MacTutor: The Indian Sulbasutras and Overview of Chinese mathematics (Zhoubi suanjing and the Gougu rule).
• Zhoubi Suanjing overview (Wikipedia), for its dating and Zhao Shuang’s 3-4-5 diagram.
• Christopher Cullen, Astronomy and Mathematics in Ancient China: The Zhou bi suan jing (Cambridge University Press, 1996), Appendix 1, pp. 206–217.
• Plimpton 322 overview (Wikipedia), for the tablet’s rows, dating and current location.

Credits: all diagrams on this page are original SVG drawings made for Jugami. The rearrangement proof is a traditional one, redrawn here.