Square Root Calculator Guide: Roots, Squares, and Estimates

If 12 squared is 144, then the square root of 144 is 12. A square root is simply the reverse of squaring: it asks which number, multiplied by itself, gives the original number. Square roots appear in geometry, in statistics, in physics, and in everyday problems such as working out the side of a square garden from its area.

The Square Root Calculator finds square roots instantly, including roots that are not whole numbers. This guide explains what a square root means, how to estimate one without a calculator, and how to check your answer.

What a square root means

Squaring a number means multiplying it by itself. Three squared is 9, because 3 times 3 is 9. Five squared is 25. Ten squared is 100. The square root reverses this: the square root of 9 is 3, the square root of 25 is 5, and the square root of 100 is 10. Every positive number has two square roots, one positive and one negative, because negative 3 times negative 3 is also 9. By convention, the square root symbol means the positive root, called the principal root, unless a problem says otherwise.

Numbers whose roots are whole numbers are called perfect squares. The first few are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144. Knowing these by heart makes estimation much faster, because any other number's root lies between two perfect squares. The square root of 50, for example, lies between 7 and 8, because 49 is 7 squared and 64 is 8 squared, and 50 is just above 49, so its root is just above 7.

Most numbers are not perfect squares, and their roots are irrational: decimals that never repeat and never end. The square root of 2 is about 1.414, but the decimal continues indefinitely. In practice, a rounded value such as 1.414 is enough, and the number of decimal places you keep should match the precision your problem needs.

Estimating a square root by hand

To estimate the root of a number that is not a perfect square, find the two perfect squares it sits between. For 50, those are 49 and 64, so the root is between 7 and 8. Then refine: since 50 is much closer to 49 than to 64, the root is much closer to 7 than to 8, perhaps around 7.1. Check by squaring: 7.1 times 7.1 equals 50.41, which is slightly above 50, so the true root is slightly below 7.1, about 7.07. One or two rounds of this guess-and-check approach usually give a good estimate.

A more systematic method is the Babylonian method, also called Heron's method. Start with a guess, divide the number by the guess, and average the guess and the result to get a better guess. Repeat until the guess stops changing much. For the square root of 50, starting with a guess of 7: divide 50 by 7 to get about 7.143, average 7 and 7.143 to get 7.0715, and squaring 7.0715 gives about 50.006, very close to 50. Two steps give an answer accurate to two decimal places.

For quick mental work, another useful fact is that the square root of a number ending in certain digits has a constrained last digit, but this matters mainly for perfect squares. For general estimation, bracketing between perfect squares and refining is enough.

Worked example: the square root of 169

Find the square root of 169. First, check whether it is a perfect square by trying nearby roots. Twelve squared is 144, and 13 squared is 169, because 13 times 13 equals 169: 10 times 13 is 130, and 3 times 13 is 39, and 130 plus 39 is 169. So the square root of 169 is 13.

Verify the answer by squaring it back: 13 times 13 is 169, which matches the original number, so the root is confirmed. As a contrast, consider the square root of 170, which is not a perfect square. It lies just above 13, because 169 is 13 squared and 196 is 14 squared. A first estimate might be 13.04, and checking gives 13.04 times 13.04 equals about 170.04, very close to 170, so the root of 170 is about 13.04.

Now apply the idea to a practical problem. A square garden has an area of 169 square metres. Its side length is the square root of 169, which is 13 metres. If the area were 170 square metres instead, the side would be about 13.04 metres, a difference of just 4 centimetres. This shows how roots translate an area back into a length, which is their most common everyday use.

Common mistakes to avoid

The most common mistake is confusing squaring with doubling. The square of 6 is 36, not 12, and the square root of 36 is 6, not 18. Another mistake is splitting a root across addition: the square root of 9 plus 16 is the square root of 25, which is 5, not the square root of 9 plus the square root of 16, which would be 3 plus 4, or 7. Roots distribute over multiplication and division, but not over addition and subtraction.

Students also sometimes forget that a negative number has no real square root. There is no real number that, multiplied by itself, gives negative 9, because a product of two numbers with the same sign is always positive. Problems that ask for the square root of a negative number are either working with complex numbers or contain an error earlier in the working.

Finally, watch your rounding. If a problem needs an answer to two decimal places, carry extra digits through the working and round only at the end. Rounding an intermediate guess can shift the final digit.

Square roots connect to many areas of maths. The Scientific Calculator computes roots and powers directly, and the Area Calculator helps with the geometry problems where roots most often appear. For a related guide on areas, see the Area Calculator Guide.

A square root is just a question: which number times itself gives this? Keep the perfect squares in mind, bracket and refine when estimating, and always check by squaring your answer. That habit turns an intimidating symbol into a routine calculation.