🎓 Mathematics • Geometry
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Pythagorean Calculator
Calculates hypotenuse or missing legs for right triangles with simplified square roots (e.g. 5√2) and decimals.
Geometry & Trigonometry Engine
Geometry, Area & Volume Studio
Solve triangles with Heron's formula and Law of Cosines, calculate circle sector arcs, compute 2D planar areas, 3D volumes, and exact Pythagorean radicals.
Target Side
Preset Right Triangles:
Solved Side (c)
5
Step-by-Step Pythagorean Proof
Formula: c² = a² + b²
Substitute: c² = (3)² + (4)²
Calculate: c² = 9 + 16 = 25
Take square root: c = √25
Exact radical form: c = 5
Decimal approximation: c ≈ 5
What is the Pythagorean Theorem?
States that the sum of the squares on the legs of a right triangle equals the square on the hypotenuse: a² + b² = c².
Formula & Step-by-Step Calculation
c = √(a² + b²), a = √(c² - b²)
Solves for the hypotenuse or any perpendicular leg.
Worked Step-by-Step Examples
Example 1
Legs 6 and 8
Solution: c = 10
• 6² + 8² = 36 + 64 = 100, √100 = 10
Common Real-World & Academic Use Cases
- ✓ Construction square corners
- ✓ Euclidean distance in game physics
- ✓ Screen diagonal sizing
How to Use the Pythagorean Calculator
1
Choose Target
Find Hypotenuse or Missing Leg.
2
Enter Dimensions
Input known side lengths.
3
View Result
Read simplified radical and decimal.
Frequently Asked Questions
Q: What is the 3-4-5 rule?
A triangle with side ratios 3:4:5 is always guaranteed to have a 90-degree right angle.