The Pythagorean theorem explained: find any side of a right triangle

A builder framing a deck measures 3 feet along one board, 4 feet along the other and checks that the diagonal between the two marks is exactly 5 feet. If it is, the corner is square. That check is the Pythagorean theorem at work, and the same rule shows your grade 8 child how to find the hypotenuse or a missing side on tonight's homework.

What does the Pythagorean theorem say?

The Pythagorean theorem says that in a right triangle, the two shorter sides squared add up to the longest side squared. OpenStax Prealgebra 2e, section 9.3 states it as "In any right triangle ΔABC, a² + b² = c²", where c is the hypotenuse and a and b are the legs.

The small raised 2 means "squared", a number times itself: 3² = 3 × 3 = 9. The guide to exponents covers that notation if it has been a while. A right triangle with legs 3 and 4 and a long side 5 fits the rule, because 9 + 16 = 25.

The picture shows why the rule is about squares. A square built on the side of length 3 holds 9 unit squares, the square on the side of length 4 holds 16, and the square on the long side holds 25. The two smaller squares together fill exactly the big one.

Which side is the hypotenuse?

The hypotenuse is the side opposite the right angle, and it is always the longest side. OpenStax defines it the same way: "The side of the triangle opposite the 90° angle is called the hypotenuse, and the other two sides are called the legs."

Find the small square corner mark first, then look across the triangle to the side it does not touch. That side is c. The legs can be called a and b in either order, because a² + b² gives the same sum both ways, and the post on types of triangles and angles shows how to spot a right angle without a protractor.

How to find the hypotenuse, step by step

How to find the hypotenuse takes four operations: square, square, add, square root. Take a right triangle with legs of 9 cm and 12 cm.

1. Name the sides. Find the right angle, mark the side across from it as c, and call the other two a and b. Here a = 9, b = 12 and c is unknown.
2. Square each leg. Multiply each leg by itself: 9 × 9 = 81 and 12 × 12 = 144.
3. Add the squares. 81 + 144 = 225. This sum is c², the area of the square on the hypotenuse.
4. Take the square root. Ask which number times itself makes 225. The answer is 15, so c = 15 cm.
5. Check the size. The hypotenuse must be longer than each leg and shorter than the two legs added together. 15 is more than 12 and less than 9 + 12 = 21, so the answer is sensible.

Step 4 is the step adults forget most often. The guide to square roots explains how to find one by hand, and knowing the perfect squares up to 15 × 15 = 225 makes most homework problems quick.

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The part people get wrong: adding the sides

The most common wrong answer adds the sides instead of the squares. A student who writes 3 + 4 = 7 for the hypotenuse of a 3-4-5 triangle has added lengths, and a slanted side can never be as long as the two legs laid end to end.

The second mistake stops at step 3. Legs of 3 and 4 give c² = 25, and 25 is the area of the square on the hypotenuse, so the side itself is 5. Say "square, add, then square root" out loud until the order is automatic.

How to find a missing leg

How to find a missing leg reverses one step: you subtract instead of adding. The hypotenuse has the biggest square, so a leg squared equals c² − b².

Take a ladder 13 feet long whose foot stands 5 feet from a wall. The ladder is the hypotenuse, so 13² = 169 and 5² = 25. Then 169 − 25 = 144, and the square root of 144 is 12, so the ladder reaches 12 feet up the wall.

The same problem can be written as an equation with one unknown. OpenStax works it as 25 + b² = 169, and solving for b uses the moves in the guide on how to solve for x: subtract 25 from both sides, then take the square root.

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Pythagorean triples worth knowing

Pythagorean triples are sets of three whole numbers that fit a² + b² = c² exactly. Five of them cover most homework sheets and building jobs.

Any whole-number multiple of a triple is also a triple. Multiplying 3, 4 and 5 by 2 gives 6, 8 and 10, and multiplying by 3 gives 9, 12 and 15, the triangle from the step list. Recognizing a triple lets you skip the square root.

Real uses: screens, ladders and square corners

The Pythagorean theorem shows up wherever two lengths meet at a right angle and you need the slanted distance between their ends. Common Core standard 8.G.B.7 asks students to "apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions."

• Screen diagonals: TV and monitor sizes are measured corner to corner. A screen 48 inches wide and 36 inches tall has a diagonal of 60 inches, because 2,304 + 1,296 = 3,600 and the square root of 3,600 is 60.
• Ladders against a wall: The wall and the ground make the right angle and the ladder is the hypotenuse. A foot 6 feet out and a top 8 feet up need a ladder 10 feet long.
• Distance on a grid: Standard 8.G.B.8 asks students to "apply the Pythagorean Theorem to find the distance between two points in a coordinate system." Walking 3 blocks east and 4 blocks north puts you 5 blocks from the start in a straight line.

The 3-4-5 corner check

The 3-4-5 corner check proves a corner is a right angle using only a tape measure. Mark 3 feet from the corner along one edge and 4 feet along the other, then measure between the two marks. A distance of exactly 5 feet means the corner is square.

The check uses the converse of the theorem: if a² + b² = c², the triangle has a right angle. Standard 8.G.B.6 asks students to "explain a proof of the Pythagorean Theorem and its converse." A diagonal longer than 5 feet means the corner is wider than 90°, and a shorter one means it is narrower. For a large frame, use 6-8-10 or 9-12-15 feet, because a bigger triangle shows small errors more clearly.

How to check progress with the Pythagorean theorem

Five short questions show whether someone understands the theorem or is copying a formula.

• Spot the hypotenuse: Show a right triangle turned on its side. The learner should point to the side across from the right angle before any calculation.
• Find a hypotenuse: Legs of 6 and 8. The answer is 10, with 36 + 64 = 100 written out.
• Find a leg: A hypotenuse of 17 and a leg of 8. The answer is 15, because 289 − 64 = 225.
• Use the converse: Do sides 5, 12 and 13 make a right triangle? Yes, because 25 + 144 = 169.
• Reject a non-right triangle: Do sides 4, 5 and 6 make a right triangle? No, because 16 + 25 = 41 and 6² = 36.

A five-minute test

A five-minute test mixes one hypotenuse, one missing leg and one converse question, with the squares written out. A learner who sets up a² + b² = c² correctly but stalls on 12 × 12 or the square root of 169 needs the square numbers more than the geometry.

Adults rebuilding their math often find that this is the real gap. The guide on how to get better at math as an adult covers rebuilding those facts in short daily sessions, and the math game for adults page shows the kind of quick-fire practice that helps.

Common Pythagorean theorem mistakes and edge cases

Four mistakes cause most wrong answers, and each has a short fix.

• Using it on a non-right triangle: The rule holds only when one angle is 90°. Check for the square corner mark, or test the sides with the converse first.
• Putting the wrong side in c: Adding the hypotenuse squared to a leg squared gives a number too big. The hypotenuse always stands alone on one side of the equals sign.
• Adding when you need to subtract: A missing leg needs c² − b². If your "leg" comes out longer than the hypotenuse, you added.
• Rounding too early: Round only the final answer. Rounding a square root halfway through and then squaring it again pushes the result off.

Edge case: answers that are not whole numbers

Many right triangles have a hypotenuse that is not a whole number. A square with sides of 1 has a diagonal of √2, because 1² + 1² = 2, and √2 is about 1.41. Legs of 2 and 3 give 4 + 9 = 13, so the hypotenuse is √13, about 3.61.

Leave the answer as √13 when the question asks for an exact value, and write "about 3.61" when it asks for a measurement. Check a rounded answer by squaring it: 3.61 × 3.61 is close to 13.

The theorem is also very old. Pythagoras lived around 500 BCE, according to OpenStax, but MacTutor's history of Babylonian mathematics describes the clay tablet Plimpton 322, dated to between 1800 BC and 1650 BC, as "a list of Pythagorean integer triples" and concludes that "the Babylonians were familiar with Pythagoras's theorem."

Your Pythagorean theorem checklist and next steps

• Right angle first: confirm one angle is 90° before using the rule.
• Find c: mark the side across from the right angle as the hypotenuse.
• Hypotenuse: square both legs, add, then take the square root.
• Missing leg: square the hypotenuse, subtract the known leg squared, then take the square root.
• Triples: look for 3-4-5, 5-12-13 and 8-15-17 and their multiples before calculating.
• Sense check: the hypotenuse is longer than each leg and shorter than the two legs added together.

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and a short daily practice that keeps square numbers and mental multiplication sharp, the facts every Pythagorean problem leans on. You can also challenge 1 to 4 friends or family members in real-time math battles, in any of 33 languages.