Pythagorean Theorem Calculator
Find the hypotenuse or any missing side of a right triangle instantly. Get step-by-step solutions, visual diagrams & unit conversion — all for free.
What is the Pythagorean Theorem?
The Pythagorean Theorem is one of the most fundamental principles in mathematics, stating that in any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Written as the equation a² + b² = c², this theorem allows you to calculate any unknown side of a right triangle when two sides are known.
Named after the ancient Greek mathematician Pythagoras of Samos (c. 570–495 BC), this theorem has been independently discovered in many cultures — including Babylonian, Indian, and Chinese mathematics — centuries before Pythagoras. Today it remains a cornerstone of geometry, trigonometry, engineering, architecture, navigation, and physics.
How to Use This Pythagorean Theorem Calculator
The Three Formulas You Need to Know
Depending on which side you're solving for, you'll use one of three rearrangements of the theorem:
- To find the hypotenuse c: c = √(a² + b²)
- To find side a: a = √(c² − b²)
- To find side b: b = √(c² − a²)
Our calculator handles all three automatically — just choose your mode and enter the known values.
Common Pythagorean Triples
Pythagorean triples are sets of three whole numbers that perfectly satisfy the theorem. These are especially useful in construction and problem-solving:
| Side a | Side b | Hypotenuse c | Verification |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 ✓ |
| 5 | 12 | 13 | 25 + 144 = 169 ✓ |
| 8 | 15 | 17 | 64 + 225 = 289 ✓ |
| 7 | 24 | 25 | 49 + 576 = 625 ✓ |
| 9 | 40 | 41 | 81 + 1600 = 1681 ✓ |
| 20 | 21 | 29 | 400 + 441 = 841 ✓ |
| 6 | 8 | 10 | 36 + 64 = 100 ✓ |
| 10 | 24 | 26 | 100 + 576 = 676 ✓ |
Real-World Applications of the Pythagorean Theorem
- Construction & Architecture: Builders use the 3-4-5 rule to ensure walls are perfectly perpendicular. It's also used to calculate roof pitch and staircase dimensions.
- Navigation & GPS: Distance between two coordinates in flat terrain is calculated using the theorem: distance = √(ฮx² + ฮy²).
- Engineering: Structural engineers calculate diagonal bracing lengths, cable tensions, and truss dimensions.
- Computer Graphics: Pixel distance calculations, collision detection, and 3D rendering all rely on the Euclidean distance formula derived from this theorem.
- Physics: Vector magnitudes, projectile motion, and wave mechanics all involve right-triangle geometry.
- Surveying & Land Measurement: Surveyors use it to measure distances across terrain, rivers, and buildings.
- Astronomy: Computing stellar distances and positions in the sky involves trigonometry rooted in the Pythagorean theorem.
Worked Examples
Example 1: Find the hypotenuse
A right triangle has legs a = 6 m and b = 8 m. Find the hypotenuse.
c = √(6² + 8²) = √(36 + 64) = √100 = 10 m
Example 2: Find a missing leg
A right triangle has a hypotenuse c = 13 cm and one leg b = 12 cm. Find side a.
a = √(c² − b²) = √(169 − 144) = √25 = 5 cm
Example 3: Diagonal of a rectangle
A rectangular room is 9 m wide and 12 m long. What is the length of the diagonal?
d = √(9² + 12²) = √(81 + 144) = √225 = 15 m
History of the Pythagorean Theorem
While Pythagoras is credited with the first formal proof around 570 BC, evidence of the theorem appears much earlier. Babylonian clay tablets from around 1800 BC list Pythagorean triples. Ancient Indian texts (the Sulba Sutras, c. 800 BC) also demonstrate knowledge of the relationship between the sides of a right triangle.
Over the centuries, more than 370 distinct proofs of the theorem have been published, including visual proofs, algebraic proofs, and even a proof contributed by US President James A. Garfield in 1876. This makes it one of the most proven theorems in mathematics.