Pythagorean Theorem Calculator
Find the missing side of any right triangle instantly — enter two sides and get the third with step-by-step solutions
Step-by-Step Solution
How to Use the Pythagorean Theorem Calculator
- Select whether you want to find the hypotenuse (c) or a missing leg (a or b) from the dropdown.
- Type the lengths of the two known sides into the input fields. Use any unit — the result will be in the same unit.
- Press the Calculate button to compute the missing side using the Pythagorean theorem formula.
- Review the detailed step-by-step solution showing the formula substitution and final answer.
- Copy the result for your work or click Clear to reset all fields and start a new calculation.
Key Features
Solve Any Side
Calculate the hypotenuse or either leg of a right triangle. Just select what you want to find and enter the two known values.
Step-by-Step Solution
Every calculation includes a full breakdown showing the formula, substitution, and intermediate steps — perfect for learning.
Instant Results
Get your answer the moment you click Calculate. No loading, no waiting — results appear immediately.
High Precision
Results are calculated with full floating-point precision and displayed up to 6 decimal places for accuracy.
Works on Any Device
Fully responsive design works on desktop, tablet, and mobile. Use it in class, on site, or at your desk.
100% Client-Side
All calculations happen in your browser. No data is sent anywhere — your inputs stay completely private.
The Pythagorean Theorem Formula
The Pythagorean theorem is one of the most fundamental relationships in geometry. It describes the connection between the three sides of a right triangle — a triangle that has exactly one 90-degree angle.
Where:
a = one leg of the right triangle (perpendicular side)
b = other leg of the right triangle (base side)
c = hypotenuse (the side opposite the right angle — always the longest side)
Depending on which side you need to find, the formula rearranges as follows:
To find the hypotenuse: c = √(a² + b²)
To find leg a: a = √(c² − b²)
To find leg b: b = √(c² − a²)
The theorem works because squaring each leg and adding them together always equals the square of the hypotenuse. This relationship holds true for every right triangle, regardless of size or proportions. The key constraint is that the triangle must contain exactly one 90° angle.
Practical Examples
🇮🇳 Rahul — Civil Engineering Student, Hyderabad
Rahul is calculating diagonal bracing for a rectangular frame. The vertical member is 3 metres and the horizontal member is 4 metres. He needs the diagonal length.
Known: a = 3 m, b = 4 m → Find c
Calculation: c = √(3² + 4²) = √(9 + 16) = √25 = 5 metres
This is the classic 3-4-5 Pythagorean triple — verified instantly.
🇮🇳 Sneha — Architect, Pune
Sneha is designing a ramp for wheelchair accessibility. The ramp's horizontal run is 12 feet and the hypotenuse (ramp surface) is 13 feet. She needs to know the rise height.
Known: b = 12 ft, c = 13 ft → Find a
Calculation: a = √(13² − 12²) = √(169 − 144) = √25 = 5 feet
The ramp rises 5 feet over a 12-foot run — a 5:12:13 Pythagorean triple.
🇺🇸 Mike — Carpenter, Austin, Texas
Mike is building a deck and needs to verify a right angle. He measures 6 feet along one edge and 8 feet along the other. He checks the diagonal to confirm it's square.
Known: a = 6 ft, b = 8 ft → Find c
Calculation: c = √(6² + 8²) = √(36 + 64) = √100 = 10 feet
If the diagonal is exactly 10 feet, the corner is a perfect 90° angle.
🇮🇳 Priya — Class 10 Student, Chennai
Priya is solving a geometry homework problem: a ladder 15 metres long leans against a wall with its base 9 metres from the wall. How high up the wall does it reach?
Known: b = 9 m, c = 15 m → Find a
Calculation: a = √(15² − 9²) = √(225 − 81) = √144 = 12 metres
The ladder reaches 12 metres up the wall — confirmed by the 9-12-15 triple (a multiple of 3-4-5).
What Is the Pythagorean Theorem?
The Pythagorean theorem is a mathematical principle attributed to the ancient Greek mathematician Pythagoras, though evidence suggests Babylonian mathematicians understood the relationship over a thousand years earlier. It states that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
This isn't just a classroom formula. It's one of the most practically applied mathematical theorems in the world. Construction workers use it daily to verify right angles (the "3-4-5 rule"). Navigation systems use it to calculate straight-line distances. Game developers use it to compute distances between objects on screen. Any time you need to find a missing dimension in a right-angled scenario, the Pythagorean theorem is your tool.
A fascinating extension is the concept of Pythagorean triples — sets of three whole numbers that perfectly satisfy a² + b² = c². The most famous triple is (3, 4, 5), but there are infinitely many, including (5, 12, 13), (8, 15, 17), (7, 24, 25), and (20, 21, 29). Any multiple of a triple is also a triple — for example, (6, 8, 10) is simply (3, 4, 5) scaled by 2.
Frequently Asked Questions
Is this tool free to use?
Yes, the Pythagorean Theorem Calculator is completely free with no usage limits, no sign-up required, and no hidden charges.
What is the Pythagorean theorem formula?
The Pythagorean theorem states that in a right triangle, a² + b² = c², where a and b are the two legs and c is the hypotenuse (the longest side opposite the right angle).
Can I find a leg if I know the hypotenuse and the other leg?
Yes. Select "Find Side a" or "Find Side b" from the dropdown, enter the hypotenuse and the known leg, and the calculator will compute the missing leg using a = √(c² − b²).
What units does this calculator use?
The calculator works with any unit — centimetres, metres, inches, feet, or any other length unit. Just ensure both input values use the same unit, and the result will be in that same unit.
Does the calculator show step-by-step solutions?
Yes. After every calculation, the tool displays a complete step-by-step breakdown showing the formula, substitution of values, intermediate calculations, and the final answer.
Can I enter decimal values?
Yes, the calculator accepts both whole numbers and decimal values. Results are displayed with up to 6 decimal places for precision.
What happens if the leg is longer than the hypotenuse?
The calculator will show an error because in a valid right triangle, the hypotenuse must always be the longest side. A leg cannot be equal to or greater than the hypotenuse.
Does the Pythagorean theorem work for all triangles?
No, the Pythagorean theorem only applies to right triangles — triangles that have exactly one 90-degree angle. For other triangles, you would use the law of cosines.
What are Pythagorean triples?
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).
Can I use this calculator for real-world construction measurements?
Absolutely. The Pythagorean theorem is widely used in construction, carpentry, surveying, and architecture to verify right angles and calculate diagonal measurements.
Recommended Hosting
Hostinger
If you are building a website for your tools, blog, or store, reliable hosting matters for speed and uptime. Hostinger is a popular option used worldwide.
Visit Hostinger →Disclosure: This is a sponsored link.
Contact Us
Have questions, suggestions, or feedback? Reach out to us anytime.