Square Root Calculator: Formula and Examples

A square root calculator finds the number that, when multiplied by itself, equals the value you enter. Enter a nonnegative number to get its principal square root, along with the square of that result as a check.

Formula

The principal square root of a nonnegative number is the nonnegative value whose square equals that number:

r = √x

• x is the number entered; it has no unit in this general calculator.
• r is the principal square root, also without a unit here.
• The check is r², which should reproduce x, apart from any displayed rounding.

For a quantity with units, the square root also changes the units: √(m²) = m and √(ft²) = ft. The calculator treats the entered value as a plain number, so interpret its units according to the original measurement.

Worked Example

Using the default input, x = 25:

1. Find the nonnegative number whose square is 25.
2. Since 5 × 5 = 25, √25 = 5.
3. Check the result: 5² = 25.

The calculator displays a square root of 5.00 and a squared-result check of 25.00.

What a Square Root Means

Squaring a number means multiplying it by itself. A square root reverses that operation: because 7² = 49, the principal square root of 49 is 7. The symbol √ is called a radical sign, and the number inside it is the radicand.

There is an important distinction between the principal square root and solving an equation. The symbol √49 means 7, not ±7. But the equation y² = 49 has two real solutions, y = 7 and y = −7, because both numbers square to 49. When solving an equation, check both signs where appropriate.

Some square roots are whole numbers, such as √64 = 8. Others are irrational and cannot be written exactly as a ratio of whole numbers. For example, √2 is approximately 1.41421356, and its decimal expansion continues without repeating. A calculator shows a rounded approximation, not the complete exact value.

How to Calculate a Square Root

For a quick calculation, enter the number into a calculator and use the √ function. On some devices, the root key is applied after entering the number; on others, it is applied to the displayed value. A result can be checked by multiplying it by itself.

For perfect squares, recognizing familiar values can make the answer immediate:

• √9 = 3, because 3² = 9.
• √100 = 10, because 10² = 100.
• √0.25 = 0.5, because 0.5² = 0.25.
• √(1/4) = 1/2, because (1/2)² = 1/4.

For a non-perfect square, estimate between nearby perfect squares. Since 6² = 36 and 7² = 49, √40 lies between 6 and 7. A numerical method or calculator can refine the estimate; to two decimal places, √40 is about 6.32.

Where Square Roots Are Useful

Square roots appear whenever a calculation reverses a squared quantity. In geometry, the diagonal of a rectangle with side lengths a and b is √(a² + b²), by the Pythagorean theorem. For sides of 3 cm and 4 cm, the diagonal is √(9 + 16) = 5 cm.

They also occur in algebra, coordinate geometry, and statistics. For example, if a square has an area of 36 square units, its side length is √36 = 6 units. The units matter: taking the root of an area in square metres gives a length in metres. In a formula involving physical measurements, ensure that quantities are expressed in compatible units before combining them.

Common Mistakes and Limits

• Confusing √x with ±√x: the radical symbol gives the nonnegative principal root. The plus-or-minus sign arises when solving an equation such as y² = x.
• Taking the root of a negative number in ordinary real arithmetic: no real number squared is negative, so √(−9) has no real value. Complex numbers extend the system, but this calculator is for real, nonnegative inputs.
• Rounding too early: keep extra digits during intermediate calculations and round the final result to the precision needed. Squaring a rounded display may not exactly recover the original input.
• Assuming roots distribute over addition: generally, √(a + b) is not √a + √b. For instance, √(9 + 16) = 5, whereas √9 + √16 = 7.
• Ignoring units: the square root of an area is a length, while the square root of a plain numerical value has no physical unit unless the input is interpreted as a measured quantity.

For very large or very small values, display precision and numerical limits can affect the shown result. The underlying value may have more digits than the screen displays, and a rounded check should be understood as approximate.

Frequently Asked Questions

What is the square root of a number?
The principal square root is the nonnegative number that, multiplied by itself, equals the given number. For example, √81 = 9 because 9² = 81.

How do I calculate a square root without a calculator?
Recognize perfect squares when possible. Otherwise, bracket the answer between nearby perfect squares, then refine the estimate with a suitable numerical method or written division method.

Can a square root be negative?
The principal square root written with √ is never negative. However, an equation such as y² = 16 has two solutions, 4 and −4.

What is the square root of zero?
It is 0, because 0 multiplied by 0 equals 0.

Why is the square root of a negative number not real?
Squaring any real number gives zero or a positive result, so no real number can have a negative square. Complex numbers provide an extended number system in which such roots can be represented.

Related

• Calculate the Hypotenuse of a Triangle
• Calculating Indefinite Integrals: Rules and Examples
• Calculating Absorbance From a Calibration Curve

in x | Number | - | 25 | 0
out root | Square root | - | sqrt(x) | 2
out check | Squared result | - | root ^ 2 | 2