Square Root Calculator

Find the square root and cube root of any number instantly. The calculator also tells you whether your number is a perfect square and shows the nearest perfect squares around it — useful for estimating roots mentally.

How to use this calculator

Enter any number — whole, decimal, even negative.
Press Calculate to see the square root and cube root.
Check the perfect-square row: it tells you whether the root is exact or irrational.

Formula used

√x = the number r where r × r = x · ∛x = the number r where r³ = x

Every positive number has two square roots (±r); by convention √x means the positive one. Negative numbers have no real square root (the result is imaginary) but do have a real cube root, since a negative × negative × negative is negative.

Example calculation

Worked example

√144 = 12, because 12 × 12 = 144 — a perfect square.

√50 ≈ 7.0711: it sits between 7² = 49 and 8² = 64, much closer to 49, so you'd estimate "just above 7" — exactly what the calculator confirms.

Square roots, explained

The square root reverses squaring: since 12² = 144, √144 = 12. Roots of non-perfect squares (like √2 or √50) are irrational — their decimals run forever without repeating — so calculators show rounded values.

To estimate any square root mentally, bracket it between perfect squares: √50 lies between √49 = 7 and √64 = 8, and since 50 is barely above 49, the answer is barely above 7. This bracketing trick is also exactly how you sanity-check a calculator result.

Why use this calculator?

Get precise roots for geometry (diagonals, areas), statistics and physics formulas.
See instantly whether a number is a perfect square.
Learn the bracketing estimates that let you approximate roots without any calculator.

Frequently asked questions

What is the square root of a negative number?

No real number squares to a negative, so the result is imaginary: √−9 = 3i, where i = √−1. The calculator shows the imaginary form. Cube roots of negatives are real, though: ∛−27 = −3.

Is √x always positive?

The symbol √x denotes the principal (positive) root by convention. But the equation x² = 25 has two solutions: +5 and −5. That distinction matters when solving equations.

How do I calculate a square root by hand?

Bracket and refine: for √50, start at 7 (since 7²=49), then average 7 and 50/7 ≈ 7.143 to get 7.071 — already accurate to three decimals. Repeating this averaging (the Babylonian method) converges very fast.