Pythagorean Theorem Calculator
Calculate the missing side of a right triangle using the Pythagorean theorem: a² + b² = c². Enter any two sides to find the third side instantly.
Find Hypotenuse (c)
Enter the lengths of both legs (a and b) to calculate the hypotenuse (c).
c = √(a² + b²)
5.00 cm
Hypotenuse length with sides a = 3 cm and b = 4 cm
Step-by-step solution:
1. Apply Pythagorean theorem: c² = a² + b²
2. Substitute values: c² = 3² + 4² = 9 + 16 = 25
3. Take square root: c = √25 = 5.00 cm
Find Side a (leg)
Enter the hypotenuse (c) and the other leg (b) to calculate side a.
a = √(c² - b²)
3.00 cm
Side a length with hypotenuse c = 5 cm and side b = 4 cm
Step-by-step solution:
1. Rearrange Pythagorean theorem: a² = c² - b²
2. Substitute values: a² = 5² - 4² = 25 - 16 = 9
3. Take square root: a = √9 = 3.00 cm
Find Side b (leg)
Enter the hypotenuse (c) and the other leg (a) to calculate side b.
b = √(c² - a²)
4.00 cm
Side b length with hypotenuse c = 5 cm and side a = 3 cm
Step-by-step solution:
1. Rearrange Pythagorean theorem: b² = c² - a²
2. Substitute values: b² = 5² - 3² = 25 - 9 = 16
3. Take square root: b = √16 = 4.00 cm
Pythagorean Theorem: Quick Framework
1. Core Formula
a² + b² = c² (c = hypotenuse)
2. Solve Patterns
- Given a,b → c = √(a² + b²)
- Given c,a → b = √(c² − a²)
- Given c,b → a = √(c² − b²)
3. Sanity Checks
- c must be largest.
- If a² + b² ≠ c² (within rounding) → input error.
- Zero or negative → invalid.
4. Triples (Fast Validation)
3‑4‑5, 5‑12‑13, 7‑24‑25, 8‑15‑17, scale by k for infinitely many.
5. Extensions
- Distance in 2D: √(Δx²+Δy²)
- Distance in 3D: √(Δx²+Δy²+Δz²)
- Trig identity: sin²θ + cos²θ = 1
6. Pitfalls
- Using diameter instead of radius in derived setups.
- Forgetting square root (reporting c²).
- Misidentifying hypotenuse (not longest side).
7. Micro Examples
6,8 → c=√(36+64)=10
c=13, b=5 → a=√(169−25)=12
8. Mini FAQ
- Decimals? Carry extra precision; round at output.
- Non-right triangle? Use law of cosines, not this.
- All three sides known? Check if right: largest² ?= sum others².
9. Action Tip
In code comparing distances, compare squared values to avoid √ cost.