Pythagorean Theorem
Solve for hypotenuse from two sides.
About the Pythagorean Theorem
For any right-angled triangle, the square of the long side equals the sum of the squares of the other two. Enter two sides and get the third.
How to use it
- Enter the two sides you know.
- Leave the unknown one blank.
- The missing side is calculated.
The formula
In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. It is the oldest theorem most people can still recite, and the most used.
a² + b² = c² so c = √(a² + b²) and a = √(c² − b²)
c is always the hypotenuse — the side opposite the right angle, and always the longest. Squaring removes any question of direction, which is why the theorem works regardless of how the triangle is oriented.
Its most common real use is not solving triangles at all: it is checking that something is square. Measure 3 units along one edge and 4 along the other, and if the diagonal between those marks is exactly 5, the corner is a true right angle. Builders call it the 3-4-5 method and use it on everything from deck frames to foundations.
Worked examples
| a | b | c (hypotenuse) | Note |
|---|---|---|---|
| 3 | 4 | 5 | the builder's triple |
| 5 | 12 | 13 | common in exam questions |
| 8 | 15 | 17 | a less familiar exact triple |
| 6 | 8 | 10 | the 3-4-5 triple doubled |
Every row here is a Pythagorean triple — three whole numbers that satisfy the theorem exactly. They are prized because they need no rounding, which is precisely why the 3-4-5 became a site standard: a carpenter can mark it with a tape and trust the result. Note the last row is simply the first multiplied by two. Scaling any triple gives another, so 9-12-15 and 30-40-50 work identically and let you pick a size that suits the job.
Common mistakes
- Solving for the hypotenuse when you have it. If you know c and one leg, you subtract rather than add: a = √(c² − b²). Adding gives a number larger than the hypotenuse, which is impossible.
- Forgetting the square root. a² + b² gives c squared, not c. Stopping one step early is the most common arithmetic slip on this theorem — 3² + 4² is 25, and the answer is 5.
- Using it on a triangle without a right angle. The theorem requires one. For other triangles use the law of cosines, which adds a correction term for the actual angle.
- Mixing units. Feet with inches, or metres with centimetres, produces a confidently wrong diagonal. Convert everything to one unit before squaring anything.
Terms explained
- Hypotenuse
- The side opposite the right angle. Always the longest side, and always c in the formula.
- Leg
- Either of the two shorter sides that meet at the right angle.
- Pythagorean triple
- Three whole numbers satisfying a² + b² = c², such as 3-4-5 or 5-12-13. Any multiple of a triple is also a triple.
- Converse
- If a² + b² = c² holds, the triangle must contain a right angle. This is what makes the theorem useful for checking square.
- Distance formula
- √[(x₂−x₁)² + (y₂−y₁)²] — Pythagoras applied to coordinates, with the horizontal and vertical gaps as the legs.
- Euclidean distance
- The straight-line distance between two points in space. The same calculation, extended to as many dimensions as you like.
Common questions
- How do I check if a corner is square?
- Measure 3 feet along one side and 4 along the other, then measure between the marks. Exactly 5 feet means a true right angle. Use 6-8-10 or 12-16-20 for larger work — the bigger the triangle, the more precise the check.
- What are the common Pythagorean triples?
- 3-4-5, 5-12-13, 8-15-17, 7-24-25 and 9-40-41, plus every multiple of each. Recognising them saves time because the answer is exact rather than a decimal.
- Can I use this on any triangle?
- Only right triangles. For others, the law of cosines generalises it: c² = a² + b² − 2ab·cos(C), which reduces to Pythagoras when C is 90°.
- How do I find a missing leg rather than the hypotenuse?
- Subtract instead of adding: a = √(c² − b²). With a hypotenuse of 13 and a leg of 5, the other leg is √(169 − 25) = 12.
- What is this used for outside geometry class?
- Checking that framing is square, finding diagonal bracing lengths, working out whether a sofa fits round a corner, computing straight-line distance between map coordinates, and measuring television screens — a screen size is its diagonal.
- Why does squaring the sides work?
- The theorem is genuinely about areas: the square built on the hypotenuse has the same area as the two squares on the legs combined. There are dozens of proofs, several of which can be done by cutting up paper.
- Does it work in three dimensions?
- Yes, by extension: the diagonal of a box is √(l² + w² + h²). You are applying the theorem twice, once for the base diagonal and once for the height.
- What is the distance formula?
- Pythagoras in disguise. The horizontal and vertical gaps between two points are the legs, and the distance between them is the hypotenuse.