Square Root Calculator

How to use this calculator

1. Enter any non-negative number.
2. The result is the principal (positive) square root, to up to 8 decimal places.

The formula

x

Any number ≥ 0

How it works

The square root reverses squaring. Because 12 × 12 = 144, √144 = 12.

Most numbers are not perfect squares; their roots are infinite non-repeating decimals that can only be approximated — this calculator rounds to 8 decimal places.

Perfect squares and irrational roots

A perfect square is the product of an integer with itself: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and onwards. Their roots are exact integers. Every other positive integer has an irrational square root — a decimal that never terminates or repeats — so any written value is necessarily a rounded approximation.

Knowing the perfect squares up to about 400 makes mental estimation quick. The square root of 60 must sit between 7 and 8 because 49 < 60 < 64, and closer to 8 because 60 is nearer 64.

Why negative numbers have no real square root

Multiplying two positive numbers gives a positive result, and multiplying two negatives also gives a positive. No real number squared can therefore produce a negative, which is why the calculator rejects negative inputs instead of returning a value.

Mathematics extends the number system with imaginary units to handle these cases, but that lies outside everyday measurement and finance work, where a negative under the root nearly always signals a data-entry error.

Where square roots appear in practice

Standard deviation is the square root of variance, which converts a squared quantity back into the original units so it can be compared with the mean. Pythagoras' theorem uses a square root to recover a length from summed squares — the basis of every diagonal and distance measurement.

In finance, volatility scales with the square root of time, so an annual figure is converted to a monthly one by dividing by the square root of twelve rather than by twelve itself.

Estimating roots by hand

Bracket the number between two perfect squares. The square root of 60 lies between 7 (49) and 8 (64), and closer to 8 since 60 is nearer 64.

Refine with one step of averaging: guess 7.7, divide 60 by 7.7 to get 7.79, then average the two for 7.745. The true value is 7.746.

This method converges very quickly and is a reliable way to check that a calculator answer is in the right neighbourhood.

Principal roots and negative numbers

Every positive number has two square roots, one positive and one negative, because both square to the same value. The principal root — the positive one — is what is reported here.

When solving an equation such as x squared = 49, both +7 and -7 are valid answers, and dropping the negative one is a common exam mistake.

Negative numbers have no real square root, since no real number squared is negative. They require imaginary numbers, which sit outside this calculator's scope.

Simplifying radicals

Pull out perfect-square factors: the square root of 72 is the square root of 36 x 2, which is 6 times the square root of 2.

Simplified radical form is exact, unlike a rounded decimal, and is the expected answer in most algebra and geometry work.

Adding radicals only works when the radical part matches: 3 root 2 plus 5 root 2 is 8 root 2, but root 2 plus root 3 cannot be combined.

Common mistakes with square roots

Assuming the root of a sum equals the sum of the roots. Root (9 + 16) is 5, not 3 + 4.

Halving instead of rooting. The square root of 36 is 6, not 18 — the two coincide only at 4.

Rounding an irrational root early and carrying it through a long calculation, which introduces drift. Keep full precision until the final step.

Practical uses of square roots

Geometry uses them constantly: the side of a square from its area, the diagonal of a rectangle via Pythagoras, and the radius of a circle from its area.

Screen and print sizing depend on them — a diagonal measurement plus an aspect ratio determines width and height through a square root.

Statistics uses them for standard deviation and standard error, and physics for free-fall times, orbital periods, and pendulum swings.

Glossary and verification

Radicand: the number under the radical sign. Principal root: the non-negative root. Perfect square: an integer whose square root is an integer. Irrational: a number that cannot be written as a fraction and whose decimal never repeats.

Verify any root by squaring it: 7.746 squared is 59.999, confirming the root of 60.

Verify a simplification by evaluating both forms as decimals — 6 root 2 and root 72 should both come to 8.485.

What a square root represents

The square root of a number is the side length of a square with that area. This geometric reading makes the operation concrete: the root of 144 is 12 because a 12 by 12 square covers 144 units.

Every positive number has two square roots, one positive and one negative, since squaring removes the sign. The principal root, the positive one, is what calculators return by default.

Negative numbers have no real square root, which is where imaginary numbers begin — useful in engineering and physics but outside everyday arithmetic.

Perfect squares, irrational roots, and rounding

Perfect squares have exact integer roots. Most numbers do not: the root of two runs forever without repeating, so any decimal you write down is an approximation.

Round only at the end of a calculation. Rounding an irrational root early and then squaring it back introduces a visible error.

Where exactness matters, keep the root in symbolic form and simplify by extracting perfect-square factors.

Estimating roots without a calculator

Bracket the number between the two nearest perfect squares. The root of 60 lies between 7 and 8, and closer to 8 because 60 is nearer 64 than 49.

Refine with one step of averaging: divide the number by your estimate and average the two. Starting from 7.7 for 60 gives 7.746 almost immediately.

This averaging method converges very fast and is worth knowing for sanity-checking any result on screen.

Where square roots appear

Pythagoras' theorem for distances and diagonals, standard deviation in statistics, quadratic solutions, and physics formulas for period and velocity.

In construction, the 3-4-5 triangle and its multiples use the same relationship to set out a true right angle.

Verify any root by squaring the answer and comparing with the original number, allowing for rounding at the last digit.

Estimating roots without a calculator

Bracketing between known squares gives a fast estimate: the root of ninety lies between nine and ten, and closer to nine and a half since ninety is near the midpoint of eighty-one and one hundred.

Refine an estimate by averaging your guess with the number divided by that guess, a method that roughly doubles the accurate digits each pass.

Two or three passes of that averaging step reach several decimal places, which is more than enough for most practical checks.

Estimating first also catches misplaced decimal points, the most common error when typing long numbers into a calculator.

Roots in geometry and statistics

The Pythagorean theorem turns two side lengths into a diagonal through a square root, which is why builders use the three-four-five triangle to check for square corners.

Standard deviation is the square root of variance, taken specifically so the spread is expressed in the same units as the data.

Doubling an area multiplies its side lengths by the square root of two, roughly 1.414, which is the basis of the A-series paper sizes.

Distance between two points on a plane is a square root of summed squares, the same relationship applied to coordinates.

Exact versus decimal answers

Most square roots are irrational, meaning their decimals never terminate or repeat, so any written decimal is an approximation.

Leaving a root in surd form keeps the answer exact, which matters when it will be squared again later in a calculation.

Simplifying a surd means extracting square factors: the root of fifty becomes five times the root of two.

Rationalising a denominator moves the root to the numerator, a convention that makes results easier to compare and to add.

When this calculator is useful

• Geometry — diagonals and the Pythagorean theorem
• Standard deviation in statistics
• Physics formulas (e.g. RMS values)
• Estimating distances