Square Root Calculator
Find the square root of any non-negative number instantly. Use it in geometry, physics, and algebra when solving for side lengths or standard deviations.
About this calculator
The square root of a number n is the value that, when multiplied by itself, gives n. Written as √n, it satisfies the equation √n × √n = n. For example, √25 = 5 because 5 × 5 = 25. Every positive number has two square roots — one positive (the principal root) and one negative — but calculators conventionally return the positive principal root. Perfect squares like 1, 4, 9, 16, and 25 yield whole-number results, while other numbers produce irrational decimals that go on forever (e.g., √2 ≈ 1.4142…). Square roots appear throughout mathematics — in the Pythagorean theorem (c = √(a² + b²)), quadratic formulas, and statistical calculations like standard deviation.
How to use
Suppose you have a square garden with an area of 144 square feet and want to know the side length. Enter 144 into the Number field. The calculator computes √144 = 12. So each side of the garden is 12 feet long. For a non-perfect square, try 50: √50 ≈ 7.071. That means a square with area 50 ft² has sides of approximately 7.07 feet. You can verify: 7.071 × 7.071 ≈ 50.
Frequently asked questions
How do I calculate the square root of a number by hand?
One common manual method is the digit-by-digit long division technique, but a simpler approach for estimates is the Newton-Raphson method: guess a starting value x, then repeatedly compute x = (x + n/x) / 2 until the result stabilizes. For √50, start with x = 7: next iteration gives (7 + 50/7) / 2 = (7 + 7.143) / 2 = 7.071. One or two iterations typically gives a result accurate to several decimal places. For perfect squares, simply memorize or test integers until you find one whose square equals your number.
Why is the square root of a negative number not a real number?
No real number multiplied by itself produces a negative result, because a positive times a positive is positive and a negative times a negative is also positive. Therefore, the square root of a negative number does not exist in the real number system. Mathematicians handle this using imaginary numbers, defining √(−1) = i, where i is the imaginary unit. Numbers involving i are called complex numbers and are used extensively in electrical engineering, signal processing, and advanced physics.
What is the difference between a square root and a perfect square?
A perfect square is an integer that is the result of squaring another integer — for example, 36 is a perfect square because 6² = 36. Its square root (√36 = 6) is therefore a whole number. Numbers that are not perfect squares still have square roots, but those roots are irrational — they cannot be expressed as simple fractions and have non-repeating, non-terminating decimals (e.g., √7 ≈ 2.6457…). Identifying perfect squares is useful in simplifying radical expressions in algebra.