Enter any non-negative number to find its square root instantly.
What is a Square Root?
In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 × 5 = 25. The symbol used for square roots is called the radical symbol: √.
Mathematically, if x2 = y, then √y = x. Every positive number has two square roots (one positive and one negative), but this calculator focuses on the "principal square root," which is the non-negative result.
Perfect Squares and Examples
A "perfect square" is a number whose square root is a whole number (an integer). Understanding perfect squares is the easiest way to estimate roots of other numbers.
Number (x)
Square Root (√x)
Calculation
1
1
1 × 1
4
2
2 × 2
9
3
3 × 3
16
4
4 × 4
25
5
5 × 5
100
10
10 × 10
How to Calculate Square Roots Manually
While this calculator provides instant results, mathematicians often use several methods to find roots manually:
Estimation: Finding the two closest perfect squares and guessing the value in between.
Newton's Method: An iterative process that starts with a guess and refines it using a specific formula: xnext = 0.5 * (x + n/x).
Prime Factorization: Breaking the number down into its prime factors to see if pairs can be extracted.
Common Uses of Square Roots
Square roots are not just for math homework; they are essential in various fields:
Geometry: Calculating the length of a side of a square when the area is known, or using the Pythagorean theorem to find the hypotenuse of a right triangle.
Physics: Formulas involving acceleration, velocity, and gravity often require square root calculations.
Statistics: Calculating the standard deviation involves taking the square root of the variance.
Engineering: Used in structural calculations and electrical engineering formulas.
function performSquareRoot() {
var inputVal = document.getElementById('radicandInput').value;
var precision = document.getElementById('precisionInput').value;
var resultDiv = document.getElementById('sqrtResultArea');
var output = document.getElementById('resultOutput');
if (inputVal === "" || inputVal === null) {
alert("Please enter a valid number.");
return;
}
var n = parseFloat(inputVal);
var p = parseInt(precision);
if (isNaN(n)) {
alert("The input must be a numeric value.");
return;
}
if (n < 0) {
resultDiv.style.display = "block";
output.innerHTML = "Error: The square root of a negative number results in an imaginary number (i). This calculator only handles real numbers.";
return;
}
var result = Math.sqrt(n);
// Format the result based on precision
var formattedResult;
if (Number.isInteger(result)) {
formattedResult = result.toString();
} else {
formattedResult = result.toFixed(p);
}
resultDiv.style.display = "block";
var htmlContent = "