Pythagorean Theorem Calculator | Free Right Triangle Tool
Find the missing side of any right triangle. Enter two sides and get the third with a visual triangle display and step-by-step solution.
What Is the Pythagorean Theorem?
The Pythagorean theorem states that in a right triangle, a² + b² = c², where a and b are the legs and c is the hypotenuse. It's one of the most fundamental relationships in geometry, named after the ancient Greek mathematician Pythagoras.
The Pythagorean Theorem Formula
The basic formula has two main variations depending on what you need to find:
Find the Hypotenuse (given both legs)
c = √(a² + b²)
Example: a=3, b=4 → c = √(9 + 16) = √25 = 5
Real-world: A 40" TV has width 34.9" and height 19.6": diagonal = √(34.9² + 19.6²) = 40"
Find a Missing Leg (given hypotenuse and one leg)
a = √(c² − b²)
Example: c=13, b=5 → a = √(169 − 25) = √144 = 12
Real-world: A 20-ft ladder placed 5 ft from a wall reaches a height of √(400 − 25) = 19.4 ft
The Distance Formula
Distance = √((x₂ − x₁)² + (y₂ − y₁)²). This is the Pythagorean theorem applied to coordinate geometry. The squared differences create a right triangle between any two points.
Real-World Applications
Construction: The 3-4-5 Rule
Builders use the 3-4-5 ratio to verify square corners. Measure 3 ft along one wall, 4 ft along the other — if the diagonal is exactly 5 ft, the corner is 90°.
Screen Sizing
TV and monitor sizes are measured diagonally. A 40" TV with width 34.9" and height 19.6" checks out: √(34.9² + 19.6²) ≈ 40".
Navigation
The shortest distance between two points on a map is a straight line — calculated using the distance formula, which is the Pythagorean theorem in disguise.
Physics: Vector Magnitude
Vector magnitude (resultant force, displacement) uses the Pythagorean theorem: |v| = √(vₓ² + vᵧ²).
Common Pythagorean Triples
Integer sets where a² + b² = c². These appear frequently because they avoid irrational numbers.
| a | b | c (hypotenuse) | Notes |
|---|---|---|---|
| 3 | 4 | 5 | Most common, used in 3-4-5 rule |
| 5 | 12 | 13 | Popular in textbook problems |
| 8 | 15 | 17 | Large legs, moderate hypotenuse |
| 7 | 24 | 25 | Often used in construction |
| 9 | 40 | 41 | Large right triangle |
| 20 | 21 | 29 | Nearly isosceles |
| 12 | 35 | 37 | Medium-sized triple |
| 11 | 60 | 61 | Extremely elongated triangle |
Any integer multiple of a triple (e.g., 6-8-10, 10-24-26) is also a Pythagorean triple.
Common Pythagorean Theorem Mistakes
Mistake 1: Using the Wrong Side as the Hypotenuse
The hypotenuse is always the longest side, opposite the right angle. If you accidentally use a leg as c, you'll get the wrong answer. Always identify the right angle first — the side across from it is c.
Mistake 2: Applying It to Non-Right Triangles
The Pythagorean theorem ONLY works for right triangles. If you try to use it on an acute or obtuse triangle, the result will be wrong. For non-right triangles, use the Law of Cosines instead.
Mistake 3: Forgetting to Take the Square Root
A common error is calculating a² + b² = c² but stopping at c². You must take the square root to get c. For a=3, b=4: c² = 25, so c = √25 = 5 — not 25.
Mistake 4: Mixing Up Legs and Hypotenuse in Subtraction
When finding a leg, you must subtract the known leg squared from the hypotenuse squared (c² − b²), not the other way around. Swapping them gives a negative number under the square root.
Limitations of the Pythagorean Theorem
The Pythagorean theorem is powerful but has important constraints. Understanding these helps you know when to use alternative methods:
- Right triangles only — It does not work for acute or obtuse triangles. Use the Law of Cosines for general triangles.
- No angle information — The theorem tells you side lengths but not angles (other than the right angle). Use trigonometry (SOH-CAH-TOA) for angles.
- Only 2D triangles — The basic formula works in two dimensions. For 3D distance, extend it: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²).
- Doesn't work in non-Euclidean geometry — On curved surfaces (like a sphere), the theorem doesn't hold. It assumes flat (Euclidean) space.
Key Takeaways
- Formula: a² + b² = c² — only works for right triangles
- c is always the hypotenuse — the longest side opposite the right angle
- Real-world: Used in construction, navigation, screen sizing, and physics
- Memorize common triples like (3,4,5) and (5,12,13) for quick problem solving
Frequently Asked Questions
What is the Pythagorean theorem formula?
a² + b² = c², where a and b are the legs (shorter sides) of a right triangle, and c is the hypotenuse (longest side, opposite the right angle).
How do I find the hypotenuse using the Pythagorean theorem?
If you know both legs (a and b), the hypotenuse is c = √(a² + b²). Example: a=3, b=4 → c = √(9 + 16) = √25 = 5.
How do I find a missing leg using the Pythagorean theorem?
If you know the hypotenuse and one leg: a = √(c² − b²). Example: c=13, b=5 → a = √(169 − 25) = √144 = 12.
What are common Pythagorean triples?
Common triples include (3,4,5), (5,12,13), (8,15,17), (7,24,25), (9,40,41), (20,21,29), (12,35,37), and (11,60,61). Any multiple of these is also a Pythagorean triple.
How is the Pythagorean theorem used in real life?
Construction (3-4-5 rule for square corners), navigation (shortest distance), screen sizing (TV diagonal), ladder placement, and the distance formula in coordinate geometry.
What is the converse of the Pythagorean theorem?
If a² + b² = c² for a triangle, then the triangle is a right triangle. This is used to check whether an angle is 90°, important in construction and surveying.
Does this work for non-right triangles?
No. The Pythagorean theorem only applies to right triangles (triangles with one 90° angle). For non-right triangles, use the Law of Cosines.
Pythagorean Theorem Calculator Pros & Cons
Pros
- Find hypotenuse or any missing leg
- Visual triangle display
- Step-by-step solution shown
- Free forever — no subscriptions
- Works offline after page load
- No sign-up or email required
Cons
- Right triangles only (not oblique)
- No angle calculation support
- No Law of Cosines for general triangles
Reviewed by Shahid
Content Reviewer & Calculator SpecialistContent reviewer specializing in marketing, finance, health, and math calculators on GM Calculator.