Pythagoras Calculator
Find the hypotenuse or any shorter side of a right-angled triangle — squared, added or subtracted, then square rooted, with full step-by-step working shown.
Pythagoras calculator
Step by step calculations
Choose what to find, enter the two known sides and press Calculate to see the step by step formula and working.
Pythagoras Calculator Explained
Core concepts, classroom applications, and plain-English explanations for all audiences.
Understanding Pythagoras' theorem
The key idea: in any right-angled triangle, the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides. Written as a formula: a² + b² = c², where c is always the hypotenuse (the side opposite the right angle).
This calculator finds any missing side — the hypotenuse, or either shorter side — as long as the triangle has a right angle and you know the other two sides.
Finding the hypotenuse
"A right-angled triangle has legs of 3 cm and 4 cm. Find the hypotenuse."
c² = a² + b²
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5 cm
The 3-4-5 triangle is the most famous Pythagorean triple — it always works and is worth memorising.
Finding a shorter side
"A right-angled triangle has a hypotenuse of 13 cm and one leg of 5 cm. Find the other leg."
a² = c² − b²
a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12 cm
Rearrange the formula: subtract the known square from the hypotenuse squared, then take the square root.
Pythagorean triples to know
These whole-number combinations always satisfy a² + b² = c²:
3 — 4 — 5 (and multiples: 6-8-10, 9-12-15)
5 — 12 — 13
8 — 15 — 17
Common Pythagorean triples
These combinations of whole numbers satisfy a² + b² = c² — worth memorising for exams.
What is the Pythagoras Calculator?
Our Pythagoras Calculator is a free online tool designed to help KS3 and GCSE students quickly find any missing side of a right-angled triangle. Whether you need to calculate the hypotenuse (the longest side) or one of the shorter sides, this calculator uses Pythagoras' theorem (a² + b² = c²) to provide accurate answers.
It's perfect for checking homework, understanding the steps involved, or solving real-world problems like calculating distances or dimensions in construction. The calculator not only gives you the answer but also shows the full step-by-step working, including squaring, adding or subtracting, and taking the square root. It even simplifies surds where possible, making complex answers easy to understand.
How to use the Pythagoras Calculator
Identify the knowns and unknowns
Look at your right-angled triangle. Which sides do you know the lengths of? Which side do you need to find? Remember, 'c' is always the hypotenuse (the longest side, opposite the right angle), and 'a' and 'b' are the two shorter sides.
Select the correct mode
Choose the mode that matches what you need to find: 'Find Hypotenuse (c)', 'Find Side a', or 'Find Side b'. This tells the calculator which formula variation to use.
Enter the known side lengths
Input the numerical values for the sides you already know into the corresponding fields. Make sure to enter positive numbers.
Click 'Calculate'
Once you've entered your values and selected the mode, click the 'Calculate' button. The calculator will instantly display the missing side length.
Review the steps and answer
The calculator will show you the full working, from squaring the sides to adding/subtracting and taking the square root. If the answer is an irrational number, it will be presented in its simplest surd form (e.g., √8 becomes 2√2).
Try these examples
Common mistakes to avoid
- Using a² + b² = c² when trying to find a shorter side:If you're finding 'a' or 'b', you need to subtract the square of the known shorter side from the square of the hypotenuse, not add. The formula becomes c² - b² = a² or c² - a² = b². When finding a shorter side, rearrange the formula: a² = c² - b² or b² = c² - a².
- Not identifying the hypotenuse correctly:The hypotenuse is always the longest side and is always opposite the right angle. If you mix up 'c' with 'a' or 'b', your calculation will be incorrect. Always identify the hypotenuse (c) first. It's the side opposite the 90-degree angle and is the longest side.
- Forgetting to take the square root at the end:Pythagoras' theorem gives you the square of the missing side (e.g., c²), not the side itself (c). Many students forget this final step. After calculating a² + b² or c² - b², remember to take the square root of the result to find the actual side length.
- Applying Pythagoras' theorem to non-right-angled triangles:Pythagoras' theorem is specifically for right-angled triangles. It will not work for acute or obtuse triangles. Only use Pythagoras' theorem for triangles that have a 90-degree angle. For other triangles, you would use the Cosine Rule or Sine Rule.