Find The Square Root Of A Number Calculator

Find the Square Root

Enter a non-negative number to calculate its square root.

Number, Root, and Square

Number (x)	Square Root (√x)	Number Squared (x²)
9	3	81

Table showing the input number, its calculated square root, and its square.

What is a Square Root Calculator?

A Square Root Calculator is a tool designed to find the square root of a given non-negative number. The square root of a number 'x' is another number 'y' which, when multiplied by itself (y × y), equals 'x'. For example, the square root of 9 is 3 because 3 × 3 = 9. Our Square Root Calculator automates this process.

This calculator is useful for students, engineers, scientists, and anyone who needs to quickly find the square root of a number without manual calculation, especially for non-perfect squares where the root is an irrational number.

Common misconceptions include thinking that negative numbers have real square roots (they have imaginary roots, which this basic Square Root Calculator doesn't handle) or that every number has an integer square root (only perfect squares do).

Square Root Formula and Mathematical Explanation

The symbol for the principal square root is √, also known as the radical sign. The expression √x means "the principal (non-negative) square root of x". So, if y = √x, then y² = x and y ≥ 0.

For example, √16 = 4 (since 4² = 16 and 4 ≥ 0). While (-4)² also equals 16, the principal square root is defined as the non-negative root.

There isn't a simple step-by-step arithmetic formula to find the exact square root of all numbers like there is for addition or multiplication, especially for irrational roots. Calculators and computers use iterative numerical methods (like the Babylonian method or Newton's method) to approximate the square root to a high degree of precision. Our Square Root Calculator uses the `Math.sqrt()` function in JavaScript, which employs such efficient algorithms.

Variables Involved

Variable	Meaning	Unit	Typical Range
x	The number whose square root is to be found (radicand)	Dimensionless (or unit²)	x ≥ 0
√x (or y)	The principal square root of x	Dimensionless (or unit)	√x ≥ 0

Practical Examples (Real-World Use Cases)

The Square Root Calculator is used in various fields.

Example 1: Geometry
You have a square-shaped garden with an area of 144 square feet and want to find the length of one side. The area of a square is side × side (side²). So, the length of a side is the square root of the area.
Input Number: 144
Using the Square Root Calculator: √144 = 12
Result: Each side of the garden is 12 feet long.

Example 2: Physics
In physics, the period 'T' of a simple pendulum is related to its length 'L' by the formula T = 2π√(L/g), where 'g' is the acceleration due to gravity. If you know T, g, and π, you might need to calculate a square root to find L or vice-versa. Or, if a distance is calculated as √(x² + y²), you'd use a square root operation.

Let's say you calculate (L/g) to be 2.25 s². To find √(L/g):
Input Number: 2.25
Using the Square Root Calculator: √2.25 = 1.5
Result: √(L/g) is 1.5 s.

How to Use This Square Root Calculator

1. Enter the Number: Type the non-negative number for which you want to find the square root into the input field labeled "Enter a non-negative number:".
2. View Results: The calculator will automatically display the square root in the "Results" section as you type or after you click "Calculate". The primary result is the calculated square root. You'll also see the original number, the root rounded, and the number squared.
3. Interpret the Chart and Table: The chart visually compares the magnitude of the number and its square root, while the table lists the number, its root, and its square.
4. Reset: Click the "Reset" button to clear the input and results and start over with the default value.
5. Copy Results: Click "Copy Results" to copy the main result and intermediate values to your clipboard.

This Square Root Calculator is straightforward. The most important output is the "Primary Result", which is the square root of the number you entered.

Key Factors That Affect Square Root Results

• Input Number's Magnitude: The larger the input number, the larger its square root, although the root grows more slowly than the number itself.
• Whether the Number is a Perfect Square: If the input is a perfect square (like 4, 9, 16, 25), the square root will be a whole number. Otherwise, it will be an irrational number (a non-repeating, non-terminating decimal), and the calculator will show an approximation.
• Non-Negativity: This basic Square Root Calculator only accepts non-negative numbers (0 or positive). The square root of a negative number is not a real number but an imaginary number (involving 'i', the square root of -1), which is outside the scope of this tool.
• Precision Required: Calculators provide an approximation of irrational roots. The number of decimal places shown depends on the calculator's internal precision and display settings.
• Computational Method: Different calculators might use slightly different algorithms or precision levels, leading to very minor differences in the last decimal places for irrational roots. Our Square Root Calculator uses standard JavaScript `Math.sqrt()`.
• Input Validity: Entering non-numeric data or leaving the field blank will result in an error or no result, as the square root is defined for numbers.

Frequently Asked Questions (FAQ)

What is the square root of 2?

The square root of 2 is an irrational number approximately equal to 1.41421356. Our Square Root Calculator can give you this value.

Can I find the square root of a negative number with this calculator?

No, this Square Root Calculator is designed for real numbers and only accepts non-negative inputs. The square root of a negative number is an imaginary number.

What is the square root of 0?

The square root of 0 is 0.

How does the Square Root Calculator handle large numbers?

It can handle large numbers within the limits of standard JavaScript number precision. Very large numbers might be displayed in scientific notation.

Is the result from the Square Root Calculator always exact?

It's exact for perfect squares. For non-perfect squares, the result is a very close approximation of the irrational square root.

What is a 'perfect square'?

A perfect square is a number that is the square of an integer (e.g., 9 = 3², 16 = 4²).

Why is the principal square root always non-negative?

By definition, the principal square root (√x) refers to the non-negative number that, when squared, equals x. This convention ensures a unique result for the √ symbol.

How is the square root used in real life?

Square roots are used in geometry (e.g., Pythagorean theorem), physics (e.g., calculating velocities, periods), statistics (standard deviation), engineering, and many other areas. Our math calculators online collection includes other useful tools.