How to Find Square Root

Finding a square root means finding the number that, multiplied by itself, gives your number. The square root of 144 is 12, because 12 × 12 = 144, and the square root of 50 is about 7.07, because 7.07 × 7.07 = 49.98, which is very close to 50. The square root calculator above gives the answer instantly, and this page teaches the hand methods: recognising perfect squares, estimating between two whole numbers, and refining the estimate by squaring.

There are three hand methods, and each suits a different situation. Perfect squares give exact answers from memory. Estimation gives a quick answer between two whole numbers. Long division, the traditional pencil-and-paper method, gives digits one at a time for any number. This page covers the first two in detail, with worked examples, and shows how to check any square root by squaring the answer.

The Square Root: What It Is and How to Find It

The square root of a number is the value that, multiplied by itself, gives the original number. The square root of 144 is 12, because 12 × 12 = 144, and the square root of 2 is about 1.41421, because 1.41421 × 1.41421 is very close to 2. The symbol is √, so √144 = 12 and √2 ≈ 1.41421. Every positive number has two square roots, one positive and one negative, but the symbol √ on its own always means the positive one, called the principal square root. The square root of a negative number is not a real number at all, which is why the calculator reports an error for negative inputs.

Most square roots are irrational numbers: their decimal expansion never ends and never repeats. The square root of 2 is the famous example — its digits run 1.41421356 and continue forever without a pattern, a fact proved by the ancient Greeks. That is why we round: √50 ≈ 7.07 is accurate to two decimal places, and √50 ≈ 7.071 is accurate to three. The calculator above also handles cube roots and other roots: the cube root of 27 is 3, because 3 × 3 × 3 = 27, and the fourth root of 81 is 3, because 3⁴ = 81.

Square roots appear wherever a square is involved: the side of a square from its area, the diagonal of a rectangle, the standard deviation in statistics, and the quadratic formula in algebra. The National Institute of Standards and Technology lists the square roots of small integers among its standard mathematical constants, and the same values turn up in geometry, physics, and engineering problems of every kind.

Step 1 — Check Whether the Number Is a Perfect Square

Try the squares you know: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. If your number is on that list, its square root is exact: √144 = 12, √225 = 15. If it is not, the root is irrational and you will estimate or use the calculator.

Step 2 — Estimate Between Two Whole Numbers

For √50, note that 49 is 7² and 64 is 8², so √50 sits just above 7. For √125, note that 121 is 11² and 144 is 12², so the answer sits just above 11, and indeed √125 ≈ 11.18. A good estimate tells you at a glance whether a calculator answer is plausible.

Step 3 — Refine and Check by Squaring

Refine the estimate by squaring your guess: 7.07² = 49.9849, just under 50, so √50 ≈ 7.07 is right to two decimal places. Always check by squaring the answer you report; if the square does not return your original number, something went wrong.

Worked Examples: Finding Square Roots

Each row shows the number, the method used, and the result, with a check by squaring where the answer is approximate.

The estimate for 50 shows the method at its best: 49 is 7² and 64 is 8², so the root sits just above 7, and 7.07² = 49.9849 confirms it to two decimal places. The square root of 2 is the most famous irrational number in mathematics: its decimal expansion never ends and never repeats, so 1.41421 is a rounded value, accurate to five decimal places, and squaring it gives 1.99999, just under 2.

How to Use the Square Root Calculator

Enter any number, choose the root, and the result appears instantly, with the exact simplified form where one exists. Use it for homework checks, geometry, and any calculation where a root is one step of a longer problem.

1. Enter the number. Type the value whose root you want, such as 72. The default of 72 shows √72 ≈ 8.49, or 6√2 in exact form.
2. Choose the root. Square root is the default; switch to cube root, fourth root, or any other root for problems such as ∛27 = 3.
3. Read the decimal result. The calculator gives the root to several decimal places, which is the form you need for measurement and engineering work.
4. Read the simplified form. Where the number has a square factor, the calculator shows the exact form: √72 = 6√2, √50 = 5√2, √125 = 5√5.
5. Square the answer to check. Multiply the result by itself and confirm you get back to the number you entered; that one check catches almost every input error.

Perfect Squares to Memorise

These squares are worth knowing by heart. They turn most square root problems into instant recall, and they anchor every estimate.

Common Mistakes With Square Roots

Square roots are a classic source of small, stubborn errors. These are the ones to watch for.

• Adding roots as if they were numbers. √2 + √3 is not √5. Roots add only when the radicand matches: √2 + 3√2 = 4√2.
• Forgetting the negative root. The equation x² = 49 has two solutions, x = 7 and x = −7, even though √49 on its own means 7.
• Taking the root of a negative number. There is no real number whose square is negative, so √(−4) has no real answer and the calculator correctly refuses it.
• Rounding too early. Rounding √2 to 1.41 and then squaring gives 1.9881, not 2. Keep extra digits until the last step of the problem.
• Confusing square and square root. Squaring makes numbers bigger than 1 larger; the square root makes them smaller. √144 = 12, while 144² = 20,736.

Where You Will Use Square Roots

Square roots are not just a school topic. They appear in practical work wherever area, distance, or variation is measured.

• Geometry. The side of a square garden with an area of 144 square metres is √144 = 12 metres, and the diagonal of a 3 by 4 rectangle is √(9 + 16) = 5.
• Statistics. The standard deviation is the square root of the variance, so every report of spread in test scores or measurements runs through a square root.
• Physics. The speed of a falling object, the period of a pendulum, and the energy in a spring all involve square roots of the quantities you measure.
• Construction. The 3-4-5 triangle, checked with √(3² + 4²) = 5, is how builders square up corners on site.
• Finance. The square root of time rule scales risk and volatility: over four times the period, typical movement roughly doubles, because √4 = 2.

Estimating Any Square Root in Three Steps

First, find the two perfect squares your number sits between: for 125, they are 121 and 144, so the root is between 11 and 12. Second, judge how close it sits to each end: 125 is only 4 above 121 and 19 below 144, so the root is just above 11, and indeed the square root of 125 is about 11.18. Third, square your estimate to check: 11.18² = 124.99, which confirms the answer to two decimal places. The same method works for large numbers: the square root of 10,000 is exactly 100, and the square root of 9,500 sits just below 100, at about 97.47, because 97.47² = 9,500.40.

Tips for Accurate Results

A little care with roots pays off, because a root error hides inside every later step of a problem.

1. Memorise the squares up to 15². Knowing 12² = 144 and 15² = 225 lets you spot perfect squares instantly and estimate everything else.
2. Simplify before approximating. Reducing √72 to 6√2 keeps the answer exact, and the decimal 8.49 follows from it without extra rounding error.
3. Keep two extra digits. Carry two more decimal places than your final answer needs, and round only at the end.
4. Check with a square. Squaring your answer is the fastest possible check, and it takes two seconds.
5. Watch the domain. The input must be zero or positive for a real square root; a negative input means the problem needs a different approach.

Square Root Quick Answers

These are the root values people look up most often, each one verified against the calculator above.

Related tools on this site: the main square root calculator, scientific calculator, basic calculator, percentage calculator, and the square root calculator again whenever you need a fresh check.

Frequently Asked Questions

How do I find the square root of a number?

Check whether it is a perfect square first: 144 is 12², so its square root is 12. If it is not, estimate between the two nearest perfect squares and refine by squaring your guess. For 50, the neighbours are 49 and 64, so the root is just above 7, and 7.07² = 49.98 confirms it.

What is the square root of 144?

The square root of 144 is 12, because 12 × 12 = 144. It is a perfect square, so the answer is exact, with no rounding and no decimal places.

How do I find the square root without a calculator?

Use the perfect squares you know, estimate between the two nearest ones, and refine by squaring. For 125, the neighbours are 121 (11²) and 144 (12²), so the root is just above 11, and 11.18² = 124.99 confirms the estimate.

What is the square root of 2?

The square root of 2 is about 1.41421. It is irrational, so its decimal expansion never ends, and 1.41421 is rounded to five decimal places. Squaring it gives 1.99999, just under 2.

Can a square root be negative?

The symbol √ always means the positive, or principal, square root, so √144 = 12. But the equation x² = 144 has two solutions, x = 12 and x = −12, because both square to 144. The square root of a negative number has no real answer.

Disclaimer: This calculator is for general information only. Results are rounded for display; for engineering, construction, or safety-critical work, confirm measurements with the applicable standards and a qualified professional.