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Free Pythagorean Theorem calculator to find hypotenuse or legs of right triangle. Calculate missing side instantly with step-by-step solutions.
Calculate missing side of right triangle using a² + b² = c²
The Pythagorean Theorem is a fundamental principle in geometry that describes the relationship between the sides of a right triangle. It states that the square of the hypotenuse (the longest side opposite the right angle) equals the sum of squares of the other two sides.
a² + b² = c²
Choose What to Find: Select whether you want to find the hypotenuse (c) or one of the legs (a or b)
Select Unit: Choose your measurement unit (mm, cm, m, inch, feet)
Enter Known Values: Input the two known side lengths of your right triangle
Calculate: Click the calculate button to get instant results
View Results: See all three sides and detailed step-by-step solution. Download or copy as needed
Formula: c = √(a² + b²)
When: You know both legs
Example: If a=3, b=4, then c = √(9+16) = √25 = 5
Formula: a = √(c² - b²)
When: You know hypotenuse and one leg
Example: If c=13, b=5, then a = √(169-25) = √144 = 12
Formula: b = √(c² - a²)
When: You know hypotenuse and one leg
Example: If c=10, a=6, then b = √(100-36) = √64 = 8
Pythagorean triples are sets of three positive integers that satisfy a² + b² = c². These are useful shortcuts for quick calculations:
3² + 4² = 5²
9 + 16 = 25 ✓
Most common triple
5² + 12² = 13²
25 + 144 = 169 ✓
Popular in construction
8² + 15² = 17²
64 + 225 = 289 ✓
Larger triple
7² + 24² = 25²
49 + 576 = 625 ✓
Another common triple
Note: Any multiple of these triples also works! For example: 6-8-10, 9-12-15, 15-20-25 are all multiples of 3-4-5.
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The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of squares of the other two sides: a² + b² = c². It's used to find the length of any side when the other two are known.
To find hypotenuse c: Use formula c = √(a² + b²). Square both legs, add them, then take square root. Example: if a=3, b=4, then c = √(9+16) = √25 = 5.
To find leg a: Use formula a = √(c² - b²). Square hypotenuse and known leg, subtract, then take square root. Example: if c=13, b=5, then a = √(169-25) = √144 = 12.
Pythagorean triples are sets of three positive integers that satisfy a² + b² = c². Common examples: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Any multiple of these also works: (6,8,10), (9,12,15).
No, Pythagorean Theorem only applies to right triangles (triangles with a 90° angle). For other triangles, use Law of Cosines or Law of Sines instead.
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This Pythagorean Theorem calculator is developed by CalcFinex, a trusted platform for accurate mathematical and financial calculators. Our tool is designed by mathematics educators and verified by geometry experts to ensure precision in every calculation.
The calculator uses the fundamental Pythagorean Theorem (a² + b² = c²), which has been mathematically proven and used for over 2,500 years. All calculations are performed using standard mathematical operations with precision up to 4 decimal places, suitable for academic, professional, and practical applications.
Accuracy & Reliability: Our calculator implements the exact mathematical formula without approximations. Results are verified against known Pythagorean triples and tested with thousands of test cases to ensure 100% accuracy.
Educational Purpose: This tool is ideal for students learning geometry, teachers creating lesson materials, engineers performing quick calculations, and professionals in construction, architecture, and design fields.