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Pythagorean Theorem Calculator
Sign up to favourite this calculatorEnter two sides of a right-angled triangle and this calculator finds the third using Pythagoras’ theorem, showing the full working plus the triangle’s area and perimeter.
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Hypotenuse (c)
The longest side, opposite the right angle
Formula: c = √(a² + b²)
Side a
3Side b
4Area
6 square unitsPerimeter
12 units| Step | Value |
|---|---|
| Pythagoras’ theorem | a² + b² = c² |
| Known side a | 3 |
| Known side b | 4 |
| Substitute the values | 3² + 4² = c² |
| Square each side | 9 + 16 = c² |
| Add the squares | 25 = c² |
| Square root | c = √25 |
| Hypotenuse | c = 5 |
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Assumptions and limits
- An estimate from the numbers you entered, for guidance only. It is not financial, tax or medical advice.
- Calculator version 1, last updated 25 October 2026.
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How to use this calculator
- Choose what to find — Hypotenuse when you know both shorter sides, or Shorter side when you know the hypotenuse and one shorter side. Only the two inputs that mode needs are shown.
- Enter the two known lengths, using the same unit for both.
- Read the result: the missing side is the large figure, with the triangle’s area and perimeter beside it and the full working in the calculation table.
Pythagoras’ theorem applies only to right-angled triangles, and the hypotenuse must always be the longest side. The calculation runs in the page as you type — nothing is sent to a server.
What do I need to enter?
- What would you like to find? — Hypotenuse (you know both shorter sides) or Shorter side (you know the hypotenuse and one shorter side). It decides which fields show.
- Side a and Side b — the two shorter sides, in hypotenuse mode.
- Hypotenuse c and Known shorter side — in shorter-side mode. The hypotenuse must be longer than the shorter side.
Every length must be greater than zero and no more than 1,000,000,000,000, in any one unit used for all sides.
What do the results show?
- Hypotenuse (c) or missing shorter side (a) — the main figure, whichever side you asked for.
- The other two sides — the sides you entered, shown beside the answer.
- Area and perimeter — of the triangle, in square units and units.
- Calculation table — the substitution, squares, sum or difference and square root, step by step.
How Pythagoras’ theorem is calculated
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
a² + b² = c²
- a and b — the two shorter sides, called the legs (interchangeable)
- c — the hypotenuse, the longest side, opposite the right angle
The same relationship rearranges to solve for whichever side is missing:
c = √(a² + b²)
- Find a shorter side: a = √(c² − b²)
- Area of the triangle: (a × b) ÷ 2
- Perimeter of the triangle: a + b + c
Results are shown to at most six decimal places with trailing zeroes removed, so an exact answer reads 5 and an irrational one reads 1.414214. No intermediate value is rounded before the final figure is formatted.
Worked example: finding the hypotenuse
A right-angled triangle has shorter sides of 3 and 4. The hypotenuse is:
| Step | Value |
|---|---|
| Pythagoras’ theorem | a² + b² = c² |
| Known side a | 3 |
| Known side b | 4 |
| Substitute the values | 3² + 4² = c² |
| Square each side | 9 + 16 = c² |
| Add the squares | 25 = c² |
| Square root | c = √25 |
| Hypotenuse | c = 5 |
The area is (3 × 4) ÷ 2 = 6 square units and the perimeter is 3 + 4 + 5 = 12 units.
Worked example: finding a shorter side
A right-angled triangle has a hypotenuse of 10 and one shorter side of 6. The missing shorter side is:
| Step | Value |
|---|---|
| Pythagoras’ theorem | a² + b² = c² |
| Known hypotenuse | c = 10 |
| Known shorter side | b = 6 |
| Rearrange the formula | a² = c² − b² |
| Substitute the values | a² = 10² − 6² |
| Square each side | a² = 100 − 36 |
| Subtract the squares | a² = 64 |
| Square root | a = √64 |
| Missing shorter side | a = 8 |
Common right-angled triangles
A Pythagorean triple is three whole numbers that satisfy a² + b² = c² exactly. Most right-angled triangles do not have whole-number sides.
| Side a | Side b | Hypotenuse c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 1 | 1 | ≈ 1.414214 |
When Pythagoras’ theorem applies
Pythagoras’ theorem works only for right-angled triangles — those with one 90° angle. The hypotenuse is always the longest side and always sits opposite the right angle. For a triangle with no right angle, use the law of cosines instead. This calculator does not check angles, so enter lengths you know belong to a right-angled triangle.
Frequently asked questions
How do I use Pythagoras’ theorem to find a missing side?
Square the two sides you know. To find the hypotenuse, add the squares and take the square root: c = √(a² + b²). To find a shorter side, subtract the smaller square from the hypotenuse squared and take the square root: a = √(c² − b²). For a = 3 and b = 4, c = √(9 + 16) = √25 = 5.
Which side is the hypotenuse?
The hypotenuse is the side opposite the right angle, and it is always the longest side of a right-angled triangle. In the formula a² + b² = c² it is c. The other two sides, a and b, are called the legs and are interchangeable.
Does Pythagoras’ theorem work for any triangle?
No. It only applies to right-angled triangles — triangles with one 90° angle. For triangles without a right angle you need the law of cosines instead. If a² + b² does not equal c² for the three sides, the triangle is not right-angled.
What units does this calculator use?
Any unit you like, as long as you use the same one for both known lengths. Enter centimetres, metres, inches or feet — the calculator never converts between units. The missing side comes out in that same unit, the area in square units and the perimeter in units.
Why must the hypotenuse be longer than the other sides?
When you know the hypotenuse and one shorter side, the missing side is √(c² − b²). If the hypotenuse equalled the shorter side the answer would be zero, and if it were shorter the value under the square root would be negative, which has no real answer. So the calculator needs c to be strictly greater than b.
How are the area and perimeter worked out?
The area of a right-angled triangle is half the product of its two shorter sides: (a × b) ÷ 2. The perimeter is the sum of all three sides: a + b + c. Both are shown once all three sides are known.
What is a Pythagorean triple?
A Pythagorean triple is a set of three whole numbers that fit a² + b² = c² exactly, such as 3-4-5, 5-12-13 and 8-15-17. Most right-angled triangles do not have whole-number sides — a triangle with legs of 1 and 1 has a hypotenuse of √2 ≈ 1.414214.
Last updated 25 October 2026.