Pythagorean Theorem Calculator
Calculate any side of a right triangle using the Pythagorean theorem a² + b² = c²
Formula Used
c = √(a² + b²) a = √(c² - b²) b = √(c² - a²)
Triangle Area
A = (a × b) / 2
Perimeter
P = a + b + c
Triangle Type
Right Triangle
90° angle present
Step 1: Square both legs
a² = ² =
b² = ² =
Step 2: Add the squares
a² + b² =
Step 3: Take the square root
c = √() =
Step 1: Square c and b
c² = ² =
b² = ² =
Step 2: Subtract b² from c²
c² - b² =
Step 3: Take the square root
a = √() =
Step 1: Square c and a
c² = ² =
a² = ² =
Step 2: Subtract a² from c²
c² - a² =
Step 3: Take the square root
b = √() =
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About Pythagorean Theorem Calculator
What is the Pythagorean Theorem?
The Pythagorean theorem is one of the most fundamental relationships in Euclidean geometry. It states that in a right triangle, the square of the length of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b).
Formula: a² + b² = c²
Where:
- a and b are the lengths of the two legs (the sides that form the right angle)
- c is the length of the hypotenuse (the side opposite the right angle)
How to Use This Calculator
- Select What to Solve For: Choose whether you want to find the hypotenuse (c) or one of the legs (a or b).
- Enter Known Values: Input the lengths of the two known sides.
- View Results: The calculator instantly shows the missing side length, plus additional measurements like area and perimeter.
Solving for Different Sides
Finding the Hypotenuse (c)
When you know both legs:
c = √(a² + b²)
Finding a Leg (a or b)
When you know the hypotenuse and one leg:
a = √(c² - b²) or b = √(c² - a²)
Additional Formulas
Triangle Area
Area = (a × b) / 2
For a right triangle, the two legs form the base and height.
Triangle Perimeter
Perimeter = a + b + c
The sum of all three sides.
Common Pythagorean Triples
Pythagorean triples are sets of three positive integers that satisfy the theorem:
| Triple | a | b | c |
|---|---|---|---|
| 3-4-5 | 3 | 4 | 5 |
| 5-12-13 | 5 | 12 | 13 |
| 8-15-17 | 8 | 15 | 17 |
| 7-24-25 | 7 | 24 | 25 |
Any multiple of these triples also works (e.g., 6-8-10 is 2× the 3-4-5 triple).
Real-World Applications
- Construction: Ensuring corners are perfectly square (3-4-5 method)
- Navigation: Calculating straight-line distances
- Architecture: Designing stairs, ramps, and roofs
- Physics: Calculating resultant vectors
- Computer Graphics: Distance calculations between points
Frequently Asked Questions
What is the hypotenuse?
The hypotenuse is the longest side of a right triangle, located opposite the 90-degree angle. It's always the side labeled 'c' in the Pythagorean theorem.
Can I use this for non-right triangles?
No, the Pythagorean theorem only applies to right triangles. For other triangles, use the Law of Cosines or Law of Sines.
What if my answer is a decimal?
Many real-world measurements don't form perfect Pythagorean triples. Decimal answers are completely valid and this calculator provides precise results.
How accurate is this calculator?
This calculator uses JavaScript's built-in Math functions, providing accuracy to approximately 15-16 significant digits.
Historical Background
While named after the Greek mathematician Pythagoras (c. 570–495 BC), this relationship was known to Babylonian mathematicians over 1,000 years earlier. The theorem appears in ancient texts from multiple cultures, including Indian and Chinese mathematics.
Note: This calculator is for educational purposes. For critical applications in engineering or construction, always verify calculations with professional tools and standards.