The Pythagorean theorem lets you find a missing side of a right triangle when you know the other two. It is one short formula, a² + b² = c², but learners often get stuck on which side is which and when to add or subtract. This guide walks through both cases with examples, the mistakes to watch for, and where the theorem shows up in everyday life.
What is the Pythagorean theorem?
The Pythagorean theorem says that in a right triangle, the square of the longest side equals the sum of the squares of the two shorter sides. Written as a formula, a² + b² = c². Here a and b are the two shorter sides, called the legs, and c is the longest side, called the hypotenuse.
Example
Take a right triangle with legs of 3 cm and 4 cm. Square both legs. 3² = 9 and 4² = 16. Add them to get 25. Now ask which number squared gives 25. The answer is 5, so the hypotenuse is 5 cm. You can check it the other way round, because 5² = 25 = 9 + 16.
Why does it work?
Picture a square drawn on each side of the triangle. The theorem says the area of the big square on the hypotenuse equals the two smaller squares together. For the 3, 4, 5 triangle, that is 25 = 9 + 16.
One way to see why is a classic picture proof. Take four copies of the same right triangle and arrange them inside a large square, so that a tilted square is left in the middle.
That middle square has side c, so its area is c². Now rearrange the same four triangles inside the same large square in a different way. The empty space becomes two squares, one with side a and one with side b. The large square and the four triangles didn't change, so the empty space didn't either. That means c² = a² + b².
How do you find the hypotenuse?
To find the hypotenuse, square the two shorter sides, add the results, and take the square root of the total. In formula form, c = √(a² + b²). The answer is always longer than either of the two shorter sides, which is a quick way to check that you haven't made a mistake.
Example
A right triangle has legs of 6 m and 8 m.
- Square each leg. 6² = 36 and 8² = 64.
- Add them. 36 + 64 = 100.
- Take the square root. √100 = 10.
The hypotenuse is 10 m. Not every answer comes out as a whole number. With legs of 5 and 7, you get 25 + 49 = 74, and √74 is about 8.60. If square roots feel slow, a few rounds of the square roots game help you recognise the perfect squares quickly. Spotting 36, 64 and 100 at a glance makes every step above faster.
How do you know which side is the hypotenuse?
The hypotenuse is always the side opposite the right angle, and it is always the longest side. Many learners get confused when the triangle is turned or drawn upside down. Find the small square that marks the right angle first. The side that doesn't touch that corner is the hypotenuse, wherever it sits on the page.
How do you find a missing shorter side?
To find a missing shorter side, square the hypotenuse, square the side you know, and subtract the smaller square from the larger one. Then take the square root. In formula form, a = √(c² − b²). You subtract here because the hypotenuse is the biggest side, so its square is the total you start from.
Example
A right triangle has a hypotenuse of 13 cm and one leg of 5 cm.
- Square the hypotenuse. 13² = 169.
- Square the known leg. 5² = 25.
- Subtract. 169 − 25 = 144.
- Take the square root. √144 = 12.
The missing side is 12 cm. Check that it makes sense. 12 is shorter than 13, as a leg should be. And 5² + 12² = 25 + 144 = 169 = 13², so the triangle fits the theorem.
The two cases follow one simple rule. If you are looking for the longest side, add. If you are looking for a shorter side, subtract. Once that rule feels natural, the geometry game is a fun place to put triangle problems into practice. It asks for quick answers, which helps the steps become automatic.
What mistakes do people make with the Pythagorean theorem?
The most common mistakes are using the theorem on a triangle without a right angle, adding when you should subtract, forgetting the square root at the end, and choosing the wrong side as the hypotenuse. Each one is easy to avoid once you know to look for it.
Example
A learner has a hypotenuse of 10 and a leg of 6, and writes 10² + 6² = 136, then √136 ≈ 11.66. That answer is longer than the hypotenuse, which is impossible for a leg. The correct step is 100 − 36 = 64, and √64 = 8.
Here are the mistakes to watch for:
- Using it on any triangle. The theorem only works when one angle is exactly 90°.
- Always calling the unknown side c. Only the hypotenuse is c. If the unknown side is a leg, you subtract.
- Stopping before the square root. 6² + 8² = 100 gives c², not c. The side is √100 = 10.
- Thinking √(a² + b²) is the same as a + b. With 3 and 4, √(9 + 16) = 5, while 3 + 4 = 7.
- Rounding too early. In problems with several steps, keep the exact value until the end and round only the final answer.
Does the Pythagorean theorem work for all triangles?
No. The Pythagorean theorem only works for right triangles, which have one angle of exactly 90°. It also works in reverse. If the three sides of a triangle fit a² + b² = c², the triangle must have a right angle. That reverse version is a reliable way to check whether a corner is square.
Example
Is a triangle with sides 5, 12 and 13 a right triangle? Square the two shorter sides and add them. 25 + 144 = 169. Square the longest side. 13² = 169. They match, so yes.
What about sides 6, 7 and 9? 36 + 49 = 85, but 9² = 81. They don't match, so the triangle has no right angle. Because 85 is bigger than 81, all its angles are smaller than 90°.
Builders use the same idea. To check that a corner is square, they measure 3 units along one wall and 4 units along the other. If the distance between those two marks is exactly 5 units, the corner is a right angle. Any unit works, such as metres or 30 cm steps, as long as all three measurements use the same one.
What is a Pythagorean triple?
A Pythagorean triple is a set of three whole numbers that fit a² + b² = c² exactly. The best known is 3, 4, 5. Others include 5, 12, 13, then 8, 15, 17 and 7, 24, 25. If you multiply every number in a triple by the same amount, you get another triple, such as 6, 8, 10.
Example
Check 8, 15, 17. 8² = 64 and 15² = 225. Together that is 289. And 17² = 289. It fits. Doubling 3, 4, 5 gives 6, 8, 10, and tripling it gives 9, 12, 15. Both of those fit too. Knowing a few triples by heart lets you spot answers instantly in many exercises.
Is the answer exact, or just close?
The answer from the Pythagorean theorem is exact, but it isn't always a whole number. A right triangle with two legs of 1 has a hypotenuse of exactly √2. As a decimal, √2 never ends, so 1.414 is only an approximation. That is why many teachers ask you to leave the answer as √2.
Example
Check the decimal version. 1.414² = 1.999396, which is close to 2 but not equal to it. The value √2 is the exact length. Write √2 when the question asks for an exact answer, and round only when it asks for a decimal, for example to two decimal places (1.41).
Where is the Pythagorean theorem used in real life?
The Pythagorean theorem is used whenever someone needs a distance they can't measure directly. Builders use it to check square corners, people use it to place a ladder safely, screen sizes are measured with it, and on a map it gives the straight-line distance between two points.
Example
A ladder is 4 m long and its foot stands 1 m from a wall. How high up the wall does it reach? The ladder is the hypotenuse, so subtract. 4² − 1² = 16 − 1 = 15, and √15 is about 3.87 m.
A few more everyday uses:
- Screen sizes. A screen that is 40 cm wide and 30 cm tall has a diagonal of √(1,600 + 900) = √2,500 = 50 cm.
- Distances on a map. Walk 3 km east and then 4 km north, and you end up 5 km from where you started, in a straight line.
- Square corners. The 3, 4, 5 check above is used for walls, garden beds and tiled floors.
Who found the Pythagorean theorem first?
The theorem is named after the Greek thinker Pythagoras, but the idea is much older. A Babylonian clay tablet known as Plimpton 322, dated to about 1800 BC, lists numbers from Pythagorean triples more than 1,000 years before Pythagoras lived, as an article on research at UNSW Sydney explains. Similar ideas appear in early Indian and Chinese mathematics.
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