What is the Pythagorean Theorem?
The Pythagorean theorem describes how the three sides of a right triangle are related. It states that the sum of the squares of the two legs equals the square of the hypotenuse:
a² + b² = c²
Where a and b are the two shorter sides (legs) of the right triangle, and c is the longest side — the hypotenuse — lying opposite to the right angle. The theorem was credited to the ancient Greek philosopher Pythagoras, who lived in the sixth century BC.
How to Use the Pythagorean Theorem Calculator
This calculator is easy to use despite its mathematical name. You simply provide any two sides of a right triangle, and the tool does the rest:
- Select your measurement system: Metric (mm, cm, m, km) or Imperial (in, ft, yd, mi).
- Choose the specific length unit you want to work with.
- Enter any two sides of the right triangle — leave the unknown side blank.
- Click Calculate. The tool will instantly display:
- The missing side (a, b, or hypotenuse c)
- Both non-right angles (α and β)
- The area of the triangle
- The perimeter of the triangle
- A step-by-step solution using the formula
Example: Ladder Against a Wall
Suppose you lean a ladder against a wall. The wall is 4 m tall and the base of the ladder is 3 m away from the wall. How long is the ladder?
- Leg a = 4 m (height of the wall)
- Leg b = 3 m (distance from wall to ladder base)
- Hypotenuse c = √(4² + 3²) = √(16 + 9) = √25 = 5 m
Enter a = 4 and b = 3 in the calculator to verify this result instantly.
Hypotenuse Formula
When you need to find the hypotenuse specifically, the Pythagorean theorem becomes the hypotenuse formula:
c = √(a² + b²)
To find a missing leg, rearrange the formula:
- Find leg a: a = √(c² − b²)
- Find leg b: b = √(c² − a²)
How to Use the Pythagorean Theorem Step by Step
Suppose you know leg a = 4 and the hypotenuse c = 8.944. To find leg b:
- Write the formula: a² + b² = c²
- Substitute: 4² + b² = 8.944²
- Calculate squares: 16 + b² = 80
- Isolate b²: b² = 80 − 16 = 64
- Take the square root: b = √64 = 8
Our calculator solves this automatically and shows the work for you.
Measurement Systems Supported
Metric Units
- mm — Millimeters (e.g., small components, engineering)
- cm — Centimeters (e.g., everyday objects, school problems)
- m — Meters (e.g., rooms, buildings, construction)
- km — Kilometers (e.g., geography, large-scale distances)
Imperial (US) Units
- in — Inches (e.g., TV screens, small woodworking)
- ft — Feet (e.g., room dimensions, US construction)
- yd — Yards (e.g., sports fields, fabric)
- mi — Miles (e.g., travel distances, geography)
Other Considerations When Dealing with Triangles
- The Pythagorean theorem applies only to right triangles (triangles with one 90° angle).
- The hypotenuse is always the longest side — it must be greater than either leg.
- The two non-right angles (α and β) always add up to 90°, since all angles in a triangle sum to 180°.
- If you only know the hypotenuse and one angle, use trigonometry (sin, cos, tan) instead.
- A 3-4-5 triangle is the most famous Pythagorean triple: 3² + 4² = 5². Other triples include 5-12-13 and 8-15-17.
Frequently Asked Questions
What is the hypotenuse?
The hypotenuse is the longest side of a right triangle. It lies directly opposite the right angle (90°). In the formula a² + b² = c², the hypotenuse is represented by c.
Can I enter the hypotenuse and one leg?
Yes! The calculator works with any two of the three sides. If you know the hypotenuse and one leg, simply enter those two values and leave the other blank. The calculator will solve for the missing leg.
Does the Pythagorean theorem work for non-right triangles?
No. The Pythagorean theorem only applies to right triangles. For other triangle types (acute, obtuse), you need the law of cosines: c² = a² + b² − 2ab·cos(C).
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers (a, b, c) that perfectly satisfy a² + b² = c². The most well-known is 3, 4, 5 — used in construction to verify right angles for thousands of years. Other examples: 5-12-13, 8-15-17, 7-24-25.
Who invented the Pythagorean theorem?
The theorem is named after the ancient Greek mathematician Pythagoras (c. 570–495 BC). However, historical evidence shows that Babylonian and Indian mathematicians knew and used the relationship centuries before Pythagoras. He (or his students) may have been the first to formally prove it.