Pythagorean theorem calculator
Find the hypotenuse or a leg of a right triangle, with area and perimeter.
Calculator inputs
Results
Enter your values and press “Calculate” to see the result.
In short
- What it calculates
- Find the hypotenuse or a leg of a right triangle, with area and perimeter.
- Formula used
a² + b² = c²- Example
- Legs of 3 and 4:
Pythagorean theorem calculator
The Pythagorean theorem links the three sides of any right triangle. Knowing two of them fixes the third.
Choose whether you are after the hypotenuse or a leg: the formula switches from a sum to a difference.
How it works
The square of the hypotenuse equals the sum of the squares of the legs. To find a leg you rearrange with a subtraction, so the hypotenuse must always be the longest side.
Formula
a² + b² = c²
Hypotenuse: c = √(a² + b²)
Leg: a = √(c² − b²)
Worked example
Legs of 3 and 4:
c = √(9 + 16) = √25 = 5
Area = 3 × 4 / 2 = 6
Perimeter = 3 + 4 + 5 = 12
Explanation
The 3-4-5 trick on site
Any builder knows how to check a square corner by measuring three units along one side, four along the other and verifying that the diagonal measures five. It is the simplest Pythagorean triple, and it remains the fastest way to square out a setting-out without instruments. Using 6-8-10 or 9-12-15 gains precision over longer distances.
Checking whether a triangle is right-angled
The theorem works both ways. If the square of the longest side equals the sum of the squares of the other two, the triangle is right-angled. If it is greater, the angle exceeds ninety degrees; if smaller, it falls short.
Distances in a plane
The distance formula between two points with coordinates is nothing but Pythagoras applied to the difference in x and y. The same idea extends to three dimensions by adding the square of the third coordinate.
Where it does not apply
It only works for right-angled triangles in plane geometry. For any triangle you need the law of cosines, which reduces to Pythagoras when the angle is right. Over the Earth's surface at long distances it does not hold either.
Frequently asked questions
Does it work for any triangle?
No, only right triangles. For a general triangle you need the law of cosines.
What are Pythagorean triples?
Sets of whole numbers that satisfy the theorem, such as 3-4-5, 5-12-13 or 8-15-17. Builders use them to set out right angles.
How do I square a corner without a set square?
With the 3-4-5 method: mark three units on one side and four on the other; if the diagonal measures exactly five, the corner is at ninety degrees.
What do I use if the triangle is not right-angled?
The law of cosines, which generalises Pythagoras by adding a term with the cosine of the included angle.
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