Square Root Calculator

What is a square root?

The square root of a number x is a number that gives x when multiplied by itself. Because 9 × 9 = 81, the square root of 81 is 9. It is written with the radical sign, √81 = 9, or with an exponent of one half: 81^1/2.

Every positive number has two square roots, one positive and one negative, since (−9) × (−9) is also 81. By convention √ always means the principal (non-negative) root, and that is the value this calculator reports. When you solve an equation such as x² = 81 you must remember both answers, x = 9 and x = −9.

Numbers like 4, 49 or 0.25 are perfect squares because their roots are rational (2, 7 and 0.5). Most other numbers have irrational roots whose decimals never end or repeat, so any calculator can only show an approximation.

How to use this calculator

1. Enter a number greater than zero in the box. Decimals are fine; use a dot as the separator.
2. Press the button. The answer appears as “Square root of … is …”.
3. Read the value: whole results are shown bare (the root of 16 is 4), others are rounded to six decimals, with shorter decimals padded to two or four places (the root of 0.25 reads 0.50).

Zero and negative numbers are rejected. A negative number has no square root among the real numbers; its roots are imaginary, for example √−9 = 3i, which this tool does not compute.

Fractions must be typed as decimals. To find √(9/16), enter 0.5625 and you get 0.75. Alternatively use √(a/b) = √a ÷ √b and take the two roots separately, 3 ÷ 4. The fraction to decimal calculator converts a fraction first if you need it.

How to find a square root by hand

Prime factorization works for perfect squares. Split the number into primes and pair them up: 1764 = 2 × 2 × 3 × 3 × 7 × 7, so the root is 2 × 3 × 7 = 42. A prime left without a partner means the number is not a perfect square.

For any other number use Newton's averaging method. Start with a guess g, then replace it with the average of g and x ÷ g, and repeat.

next guess = (g + x ÷ g) ÷ 2Each round roughly doubles the number of correct digits.

Example: find √50 starting from 7. Round one gives (7 + 50 ÷ 7) ÷ 2 = 7.071429. Round two gives (7.071429 + 50 ÷ 7.071429) ÷ 2 = 7.071068. The calculator confirms √50 = 7.071068.

Rules for working with square roots

• Products split: √(a × b) = √a × √b, so √72 = √36 × √2 = 6√2.
• Quotients split: √(a ÷ b) = √a ÷ √b.
• Squaring undoes the root for non-negative values: (√a)² = a, while √(a²) = |a|.
• Sums do not split: √(9 + 16) = 5, but √9 + √16 = 7.
• A root is a power: √x = x^1/2, which connects it to the exponent calculator and the root calculator.

Square roots worth knowing

The first block lists perfect squares, the second the roots of small non-squares, rounded to six decimals.

Where square roots show up

• Geometry: the side of a square with area A is √A, so a 72.25 m² plot is 8.5 m on a side. The hypotenuse of a right triangle with legs 3.6 and 4.8 is √(3.6² + 4.8²) = 6. The area calculator goes the other way.
• Statistics: a standard deviation is the square root of a variance, as in the standard deviation calculator. A variance of 6.25 means a spread of 2.5.
• Physics: formulas for wave speed, pendulum period and root-mean-square values all contain a square root.
• Finance: volatility is scaled with the square root of time, so a daily figure is multiplied by about √252 to estimate a yearly one.

Common mistakes

• Treating √(x² + y²) as x + y. It is not; for 3 and 4 the left side is 5 and the right side is 7.
• Forgetting the negative root when solving x² = k.
• Rounding too early. Keep extra digits through a calculation and round once at the end.
• Expecting a real answer for a negative input.