Square Root Calculator
Last updated: · ISO 80000-2: Mathematical Signs and Symbols
√n = n^(1/2)
Formula
√n = n^(1/2)About This Calculator
You are laying out a square garden bed and need to know the side length from the area. Or you are working through the Pythagorean theorem to find the hypotenuse of a right triangle. Or a physics problem gives you the square of a velocity and asks for the speed. Square roots appear constantly in geometry, engineering, physics, and statistics — any time you work backwards from an area or a squared value to a length or original number. This calculator gives you the precise square root instantly, along with the cube root and the square for full reference. Enter your number, click Calculate, and get the result to 8 decimal places.
How It Works
The square root of a number n is the value that, when multiplied by itself, equals n. Written as √n or n^(1/2). For example, √25 = 5 because 5 × 5 = 25. Perfect squares (1, 4, 9, 16, 25...) have integer square roots. All other positive numbers have irrational square roots — they cannot be expressed as a simple fraction and their decimal expansion never terminates or repeats. The calculator uses JavaScript's Math.sqrt() function, which implements IEEE 754 double-precision floating-point arithmetic per the ISO/IEC 10967 standard.
Step-by-Step Examples
Finding the Side of a Square from Its Area
You have a square room with a floor area of 225 square feet.
- 1.Enter n: 225
- 2.Formula: √225 = ?
- 3.15 × 15 = 225, so √225 = 15
- 4.The room is exactly 15 feet on each side
Applying the Pythagorean Theorem
A right triangle has legs of 6 cm and 8 cm. Find the hypotenuse.
- 1.Formula: c = √(a² + b²)
- 2.c = √(6² + 8²) = √(36 + 64) = √100
- 3.Enter n: 100
- 4.√100 = 10 cm
Quick Reference Chart
| Number (n) | Square Root (√n) |
|---|---|
| 1 | 1 |
| 2 | 1.414 |
| 3 | 1.732 |
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
| 25 | 5 |
| 50 | 7.071 |
| 100 | 10 |
| 144 | 12 |
Common Mistakes to Avoid
Confusing √(a + b) with √a + √b
√(a + b) ≠ √a + √b. For example, √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. Always calculate the sum first, then take the square root.
Impact: This error appears frequently in the Pythagorean theorem and leads to incorrect hypotenuse calculations.
Expecting a whole number answer for every input
Only perfect squares (1, 4, 9, 16, 25, 36...) have integer square roots. Most numbers produce irrational results that must be expressed as decimals.
Real-World Applications
Construction & Architecture
Builders calculate diagonal measurements, staircase rise/run ratios, and roof pitch using square roots. The Pythagorean theorem is fundamental to ensuring right angles in any structure.
Physics & Engineering
Speed from kinetic energy, voltage from power, and RMS values in electrical engineering all require square roots. Any formula involving squared quantities uses √ to solve for the original variable.
Finance & Statistics
Standard deviation is the square root of variance. Portfolio volatility, risk metrics, and statistical confidence intervals all involve square roots. Financial analysts use them daily.
Frequently Asked Questions
What is the square root of 2?
√2 = 1.41421356... — an irrational number that continues infinitely without repeating. It represents the diagonal length of a square with side 1, and appears frequently in geometry and trigonometry.
How do I calculate square root by hand?
The simplest manual method is the "guess and check" approach: estimate √n, square your guess, adjust up or down, repeat. For √50: try 7 → 7² = 49 (close), try 7.07 → 7.07² = 49.98 (very close). The precise answer is 7.07107.
What is the square root of a negative number?
Square roots of negative numbers are not real numbers — they are imaginary numbers involving i, where i = √(−1). For example, √(−9) = 3i. Real-world physical calculations typically require real (non-negative) inputs.
Is √4 equal to 2 or −2?
By mathematical convention, the square root symbol (√) denotes the principal (positive) root. So √4 = 2, not −2, even though (−2)² = 4. When solving x² = 4, both x = 2 and x = −2 are solutions, but √4 by itself always refers to the positive root.
What numbers have perfect square roots?
Perfect squares are: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225... These are squares of integers (0², 1², 2², 3²...). Any number not in this sequence has an irrational square root.