Pythagorean Theorem Calculator
Find the hypotenuse or a missing side of a right triangle, or check whether three sides make one. Answers are exact, like 5√2, with the decimal too, and the squares on the sides are drawn to scale.
Any unit works, as long as every side uses the same one. Decimals (2.5) and fractions (3/4) are fine.
Type the sides in any order: the longest one is treated as the hypotenuse.
Hypotenuse
c = 5
With legs 3 and 4, the hypotenuse is c = 5.
- Area
- 6
- Perimeter
- 12
- Angles
- 36.87°, 53.13° and 90°
Show me how
Why does this work?
All steps as text
Tip: press ← and → to move between steps, or click any step.
The Pythagorean theorem
In a right triangle, the square on the longest side, the hypotenuse c, is as big as the squares on the two shorter sides (the legs) together:
With legs of 3 and 4, the squares are 9 and 16. Together they make 25, the square on the hypotenuse, so c = √25 = 5.
How to find the hypotenuse
- Square the two legs: 5² = 25 and 12² = 144.
- Add them: 25 + 144 = 169.
- Take the square root: c = √169 = 13.
When the sum isn’t a perfect square, the answer is a square root. Take out any square factor to write it exactly: with legs 5 and 5, c = √50 = √(25 × 2) = 5√2 ≈ 7.0711.
How to find a missing leg
Turn the formula round: b² = c² − a². With a hypotenuse of 13 and one leg of 5, b² = 169 − 25 = 144, so b = 12. The hypotenuse must be the longest side; if it isn’t, there is no such triangle.
Is it a right triangle?
Square all three sides and compare the two smaller squares with the biggest. For 6, 8 and 10: 36 + 64 = 100 = 10², so it is a right triangle. If a² + b² is bigger than c², every angle is less than 90° (an acute triangle); if it is smaller, the angle opposite c is more than 90° (an obtuse triangle).
Pythagorean triples
Whole numbers that fit a² + b² = c² exactly are called Pythagorean triples. Any multiple of a triple is another one: 3, 4, 5 gives 6, 8, 10 and 9, 12, 15.
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
| 9 | 40 | 41 |
Two special right triangles
- 45-45-90: the two legs are equal and the hypotenuse is √2 times as long. Legs of 1 give a hypotenuse of √2 ≈ 1.4142, half a square cut along its diagonal.
- 30-60-90: the sides are in the ratio 1 : √3 : 2. A hypotenuse of 2 and a short leg of 1 give a long leg of √3 ≈ 1.7321, half an equilateral triangle.
Where it is used
- Screens: a 55-inch TV is measured along the diagonal. With a 16:9 screen, it is about 47.9 inches wide and 27.0 inches tall.
- Ladders: a 5 m ladder with its foot 1.4 m from a wall reaches √(5² − 1.4²) = √23.04 = 4.8 m up it.
- Square corners: builders mark 3 units along one wall and 4 along the other. If the diagonal is exactly 5, the corner is square. 6, 8, 10 works the same for a bigger check.
Questions
What is the hypotenuse?
The longest side of a right triangle, opposite the right angle. The other two sides, which meet at the right angle, are the legs.
Does it work for any triangle?
No, only for right triangles. For other triangles, the law of cosines adds a correction: c² = a² + b² − 2ab cos C, which becomes Pythagoras when C is 90°.
Why is the answer written as 5√2?
√50 isn’t a whole number, and its decimal goes on forever without repeating. 5√2 is its exact value, which is what a maths class usually wants; 7.0711 is the rounded decimal, for measuring.
Why is the theorem true?
One classic proof: four copies of the triangle fit inside a square of side a + b, leaving a tilted square of side c in the middle. Rearranged, the same four triangles leave two squares, a² and b². The empty space is the same both times, so c² = a² + b².
Can I use decimals, fractions or other units?
Yes: 2.5, 3/4 and lengths in centimetres, inches or feet all work, as long as every side uses the same unit. The answer is in that unit too.
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