Math

Square Root Calculator

Calculate the square root of any positive number instantly. See step-by-step approximations and learn about perfect squares.

Enter a Number

√144

Common Square Roots

√1 = 1 √4 = 2 √9 = 3 √16 = 4 √25 = 5 √100 = 10 √144 = 12 √225 = 15 √400 = 20 √10000 = 100

Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, equals the original number. For example, √144 = 12 because 12 × 12 = 144.

Perfect Squares

A perfect square is a number whose square root is a whole number. Here are the first 20 perfect squares:

Number (n) n² √n²
1 1 1
2 4 2
3 9 3
4 16 4
5 25 5
6 36 6
7 49 7
8 64 8
9 81 9
10 100 10
11 121 11
12 144 12
13 169 13
14 196 14
15 225 15
16 256 16
17 289 17
18 324 18
19 361 19
20 400 20

Square Root Properties

  • √(a × b) = √a × √b — The square root of a product equals the product of square roots
  • √(a ÷ b) = √a ÷ √b — The square root of a quotient equals the quotient of square roots
  • √(a²) = a — The square root of a squared number equals the base number
  • √ab = √a × √b — Useful for simplifying radicals

Common Uses

  • Geometry: Finding the side of a square given its area
  • Pythagorean Theorem: a² + b² = c², so c = √(a² + b²)
  • Statistics: Standard deviation calculations
  • Physics: Velocity and acceleration formulas
  • Engineering: Signal processing and control systems

Frequently Asked Questions

What's the square root of 0?

The square root of 0 is 0, because 0 × 0 = 0. Zero is unique as both a perfect square and having only one square root.

Can I calculate negative square roots?

Every positive number has two square roots: one positive and one negative. For example, √144 = ±12. Our calculator shows the principal (positive) square root.

What's an irrational square root?

Most square roots are irrational numbers (they go on forever without repeating). √2 ≈ 1.41421356... is the most famous example. These cannot be expressed as exact fractions.

How accurate is the calculation?

Our calculator provides results accurate to 8 decimal places, which is more than sufficient for any practical application. The system uses PHP's built-in sqrt() function for precision.