Square Root Calculator
Calculate the square root of any number instantly. Our calculator shows results with adjustable precision and identifies perfect squares.
Result
Quick Examples (Perfect Squares 1-20)
What Is a Square Root?
The square root of a number is a value that, when multiplied by itself, gives the original number. For example, √16 = 4 because 4 × 4 = 16. The symbol √ is called a radical, and the number under it is called the radicand.
√x = y means y × y = x
Types of Square Roots
Perfect Squares
Numbers whose square roots are whole numbers: 1, 4, 9, 16, 25...
Irrational Roots
Non-perfect squares have infinite, non-repeating decimals: √2 = 1.41421...
Imaginary Roots
Negative numbers have imaginary square roots: √-1 = i
Properties of Square Roots
Product: √(a × b) = √a × √b
Quotient: √(a / b) = √a / √b
Power: √(a²) = |a|
Perfect Squares Reference Table (1–25)
A perfect square is a whole number whose square root is also a whole number. Memorizing these speeds up mental math, algebra, and standardized test problems.
| n | n² (Perfect Square) | √n² = n |
|---|---|---|
| 1 | 1 | √1 = 1 |
| 2 | 4 | √4 = 2 |
| 3 | 9 | √9 = 3 |
| 4 | 16 | √16 = 4 |
| 5 | 25 | √25 = 5 |
| 6 | 36 | √36 = 6 |
| 7 | 49 | √49 = 7 |
| 8 | 64 | √64 = 8 |
| 9 | 81 | √81 = 9 |
| 10 | 100 | √100 = 10 |
| 11 | 121 | √121 = 11 |
| 12 | 144 | √144 = 12 |
| 13 | 169 | √169 = 13 |
| 14 | 196 | √196 = 14 |
| 15 | 225 | √225 = 15 |
| 16 | 256 | √256 = 16 |
| 17 | 289 | √289 = 17 |
| 18 | 324 | √324 = 18 |
| 19 | 361 | √361 = 19 |
| 20 | 400 | √400 = 20 |
| 21 | 441 | √441 = 21 |
| 22 | 484 | √484 = 22 |
| 23 | 529 | √529 = 23 |
| 24 | 576 | √576 = 24 |
| 25 | 625 | √625 = 25 |
How to Estimate Square Roots by Hand
For non-perfect squares, use the Babylonian method (Newton's method) — it converges to the correct answer in just a few steps.
Algorithm:
1. Start with an initial guess g (any reasonable number)
2. New guess = (g + n / g) / 2
3. Repeat step 2 until the result stabilizes
Example: √20
Start: 4² = 16, 5² = 25 → guess g = 5
Step 1: (5 + 20/5) / 2 = (5 + 4) / 2 = 4.5
Step 2: (4.5 + 20/4.5) / 2 ≈ 4.472
Step 3: (4.472 + 20/4.472) / 2 ≈ 4.4721
Result: √20 ≈ 4.4721 ✓ (actual: 4.47213…)
Real-World Applications of Square Roots
Geometry & Construction
The Pythagorean theorem (a² + b² = c²) uses a square root to find a side: c = √(a² + b²). Essential for construction, carpentry, and navigation.
Area Calculations
To find the side length of a square from its area, take the square root. A 196 m² garden has sides of √196 = 14 m.
Finance: Volatility & Returns
Standard deviation — a key investment risk measure — involves square roots. Annualizing monthly volatility requires multiplying by √12.
Physics & Engineering
Speed, frequency, and force formulas regularly involve square roots. The period of a pendulum is T = 2π × √(L/g).