What the square root calculator does
Enter a number and the calculator returns its square root, its cube root, and whether it is a perfect square. Square roots appear everywhere a distance, a standard deviation, an area's side length or a quadratic solution is needed; cube roots appear in volumes and growth rates. The result is computed in your browser to full floating-point precision.
Definitions
√n = the number that, squared, gives n √144 = 12 because 12² = 144
³√n = the number that, cubed, gives n ³√27 = 3 because 3³ = 27
√n = n^(1/2) ³√n = n^(1/3) (any root is a fractional power)Every positive number has two square roots (12 and −12 both square to 144); the calculator gives the principal, positive root, which is what the √ symbol means. Cube roots of negative numbers are real and negative: ³√(−8) = −2.
Perfect squares and cubes
| n | √n | n | √n | n | ³√n |
|---|---|---|---|---|---|
| 1 | 1 | 121 | 11 | 1 | 1 |
| 4 | 2 | 144 | 12 | 8 | 2 |
| 9 | 3 | 169 | 13 | 27 | 3 |
| 16 | 4 | 196 | 14 | 64 | 4 |
| 25 | 5 | 225 | 15 | 125 | 5 |
| 36 | 6 | 256 | 16 | 216 | 6 |
| 49 | 7 | 400 | 20 | 343 | 7 |
| 64 | 8 | 625 | 25 | 512 | 8 |
| 81 | 9 | 900 | 30 | 729 | 9 |
| 100 | 10 | 10000 | 100 | 1000 | 10 |
A number is a perfect square if its square root is a whole number. Recognising them speeds up mental arithmetic and simplifying radicals: √50 = √(25 × 2) = 5√2.
Estimating a square root by hand
Guess-and-improve (Newton's method):
next guess = (guess + n ÷ guess) ÷ 2
√30: guess 5 → (5 + 6) ÷ 2 = 5.5 → (5.5 + 5.4545) ÷ 2 = 5.4773 → 5.4772…
Two rounds already give four correct digits.For a quick mental estimate, find the nearest perfect squares: √30 is between √25 = 5 and √36 = 6, closer to 5 — about 5.5.
Where square roots show up
- Distance: the straight-line distance between two points is √(Δx² + Δy²) — Pythagoras.
- Statistics: standard deviation is the square root of variance; standard error shrinks with √n.
- Geometry: the side of a square with area A is √A; the diagonal of a square is side × √2.
- Finance: annualising volatility multiplies by √(periods per year); the CAGR over n years is the n-th root of the growth factor.
- Physics: the period of a pendulum, the speed of a wave, and escape velocity all involve square roots.
- Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a.