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Geometry & Measurement

Pythagorean Theorem Calculator

Enter any two sides of a right triangle and leave the third blank. The calculator picks the right rearrangement of a² + b² = c², shows each substitution, and checks the result. Fill all three to test whether a triangle is right-angled.

Updated October 3, 2026Written and reviewed by Mateuss M.Method: how we calculate

Try an example:

Answerc = 5

Area of the triangle: 6
Perimeter: 12

Step-by-step working 4 steps

1
The theorem
```
a² + b² = c²
```
a and b are the two shorter sides (legs); c is the hypotenuse, opposite the right angle.
2
Square the legs and add
```
3² + 4² = 9 + 16 = 25
```
3
Take the square root
```
c = √25 = 5
```
4
Check
```
3² + 4² = 25  and  5² = 25
```

How to use the pythagorean theorem calculator

Enter the two sides you know.
Leave the unknown side blank.
Press Solve and read the substitution steps.
Enter all three sides to test whether they make a right triangle.

The Pythagorean theorem

a² + b² = c²
c = √(a² + b²)
a = √(c² − b²)

It only applies to right triangles, and c is always the longest side.

Worked examples

Each example below is generated by the same calculator you used above, so the working always matches the answer.

Legs 3 and 4

Answerc = 5

Area of the triangle: 6
Perimeter: 12

Step-by-step working 4 steps

1
The theorem
```
a² + b² = c²
```
a and b are the two shorter sides (legs); c is the hypotenuse, opposite the right angle.
2
Square the legs and add
```
3² + 4² = 9 + 16 = 25
```
3
Take the square root
```
c = √25 = 5
```
4
Check
```
3² + 4² = 25  and  5² = 25
```

Hypotenuse 13, leg 12

Answera = 5

Area of the triangle: 30
Perimeter: 30

Step-by-step working 4 steps

1
The theorem
```
a² + b² = c²
```
a and b are the two shorter sides (legs); c is the hypotenuse, opposite the right angle.

2

Rearrange for the missing leg

a² = c² − b²
a² = 13² − 12² = 169 − 144 = 25

3
Take the square root
```
a = √25 = 5
```
4
Check
```
5² + 12² = 169  and  13² = 169
```

Legs 5 and 7

Answerc = 8.602325

Area of the triangle: 17.5
Perimeter: 20.6023

Step-by-step working 4 steps

1
The theorem
```
a² + b² = c²
```
a and b are the two shorter sides (legs); c is the hypotenuse, opposite the right angle.
2
Square the legs and add
```
5² + 7² = 25 + 49 = 74
```
3
Take the square root
```
c = √74 = 8.602325
```
4
Check
```
5² + 7² = 74  and  8.6023² = 74
```

Test 6, 8, 10

AnswerYes — a right triangle

Step-by-step working 3 steps

1
The theorem
```
a² + b² = c²
```
a and b are the two shorter sides (legs); c is the hypotenuse, opposite the right angle.

2

Test the sides

a² + b² = 6² + 8² = 100
c² = 10² = 100

3
Compare
Both sides are equal, so the sides satisfy the theorem.

Common mistakes to avoid

Treating the hypotenuse as a leg

c must be the longest side. When solving for a leg you subtract squares; adding them would give a length longer than the hypotenuse.

Forgetting the final square root

a² + b² gives c², not c. Take the square root at the end.

Using it on a non-right triangle

The theorem only holds when one angle is exactly 90°.

Frequently asked questions

What are Pythagorean triples?

Sets of whole numbers that satisfy the theorem, such as 3-4-5, 5-12-13 and 8-15-17. Multiples of a triple also work, like 6-8-10.

Can I use decimals?

Yes. Any positive numbers work; results are shown to six decimal places.