Square Root Calculator

About this Square Root Calculator

The Square Root Calculator returns the principal square root of a number, meaning the non-negative value that multiplies by itself to produce the original input. It is useful for geometry, algebra and quick numeric checks.

The square root operation reverses squaring, and it appears throughout geometry, physics and statistics, making it one of the more frequently needed operations beyond basic arithmetic.

For related tasks, try Exponent Calculator, Modulo Calculator or Average Calculator.

How the calculation works

The browser calculates the principal real square root. Negative inputs do not have a real-number square root, so this tool does not return complex values.

How to use it

1. Enter a non-negative number.
2. Select Calculate square root.
3. Review the square root result.

Practical example

The square root of 144 is 12 because 12 × 12 = 144.

The square root of 2 is an irrational number, approximately 1.41421356, meaning its decimal digits continue forever without repeating, unlike the square root of a perfect square such as 144, which is exactly 12.

Things to double-check

Every positive number technically has two square roots, one positive and one negative, since both, when squared, produce the same positive result; this calculator returns the principal (positive) square root, which is the convention used in most everyday and academic contexts.

Understanding square roots

The square root of a number is the value that multiplied by itself gives that number. The square root of 144 is 12, because 12 times 12 is 144. Numbers that are not perfect squares have irrational roots with endless decimals: the square root of 50 is about 7.0711, which lies between 7, whose square is 49, and 8, whose square is 64. A quick estimate by finding the neighbouring perfect squares helps you check the answer.

Simplifying roots

A root can often be written in a simpler exact form. The square root of 50 equals the square root of 25 times 2, which is 5 times the square root of 2. The square root of 72 is 6 times the square root of 2, since 72 is 36 times 2. Exact forms are used in algebra and geometry because they avoid rounding. The decimal from the calculator is useful for measurements, and the simplified form is better for exact work.

Geometry and measurement

Square roots appear whenever you convert between area and length. A square plot of 250 square metres has sides of about 15.81 m. The Pythagorean theorem uses them as well: the diagonal of a 3 by 4 rectangle is the square root of 25, which is 5, and the diagonal of a 10 by 10 square is the square root of 200, about 14.14. In statistics, the standard deviation is the square root of the variance.

Estimating by hand

You can improve an estimate without a calculator. To find the square root of 20, guess 4.5. Divide 20 by 4.5 to get about 4.444, then average 4.5 and 4.444 to get 4.472, which is already very close to the true value of 4.4721. Repeating this step improves the answer further. The tool only accepts non-negative numbers because the square root of a negative number is not a real number.

Distance and the Pythagorean theorem

Square roots give straight-line distances. The distance between the points (1, 2) and (4, 6) is the square root of 3 squared plus 4 squared, which is the square root of 25, or 5. The same idea gives the diagonal of a screen or a room: a rectangle 1.2 m by 0.5 m has a diagonal of 1.3 m. Measuring a diagonal is also a practical way to check that a frame or foundation is square, because both diagonals of a true rectangle are equal.

Worked example: a square garden plot

A square plot has an area of 180 square metres. Its side is the square root of 180, which is about 13.42 metres, and the perimeter is four times that, about 53.7 metres. That is the length of fence you need. A quick check: 13 squared is 169 and 14 squared is 196, so the root must lie between 13 and 14, and 13.42 fits. If the plot were a rectangle with sides in the ratio 2 to 1, you would set up the area as 2 times the shorter side squared, which gives a different shape. The square root is the tool that turns an area back into a length.

Irrational roots and a method to find them

Most square roots are irrational numbers, which means their decimals never end or repeat. The square root of 2 is about 1.41421356, of 3 about 1.73205 and of 5 about 2.23607. A simple way to calculate a root by hand is the Babylonian method: start with a guess and average it with the number divided by the guess. For the square root of 50, start with 7. The average of 7 and 50 over 7, which is 7.1429, is 7.0714. Repeating gives 7.07107, which is correct to five decimal places. Each step roughly doubles the number of correct digits.

Applications in science and geometry

Square roots appear in many formulas. The distance formula and the Pythagorean theorem both use them. In physics, the time for an object to fall a distance h from rest is the square root of 2h divided by g, so a fall of 20 metres takes about the square root of 4.08, which is 2.02 seconds. In statistics, the standard deviation is the square root of the variance, and in finance, volatility scales with the square root of time. When a formula contains a square root, check that the number inside it is not negative, since the result would not be a real number.

Exact and decimal answers

In school mathematics, an answer such as 5 times the square root of 2 is exact and often preferred, while the decimal 7.0711 is an approximation. For measuring and building, use decimals and round sensibly: a length of 13.42 metres is fine for fencing, but not for machining a part. Remember that the square root of a product equals the product of the square roots, so the square root of 72 is the square root of 36 times 2, which is 6 times the square root of 2. The square root of a sum, however, is not the sum of the roots, which is a frequent mistake.

Key terms explained

A square is a number multiplied by itself. The square root is the number that gives the original when squared. A perfect square has a whole-number root, and the radicand is the number under the root sign. An irrational root has a decimal that never ends or repeats. The principal root is the positive one. A cube root is the number that gives the original when multiplied by itself twice, and the hypotenuse of a right triangle can be found with a square root.

Rules of thumb and quick checks

Learn the perfect squares up to 20: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361 and 400. The root of a product equals the product of the roots, so the root of 72 is 6 times the root of 2. To estimate, find the perfect squares on either side. A perfect square never ends in 2, 3, 7 or 8. If the radicand is negative, the result is not a real number.

Frequently asked questions