How to Calculate a Square Root by Hand
How to calculate a square root by hand with two verified methods: the averaging (Babylonian) method with three full worked examples, and the classic long-division algorithm, digit by digit.
A calculator finds a square root in an instant, but knowing how to calculate a square root by hand is still worth learning, for exams that restrict calculators, for building real number sense, and honestly, for understanding what a calculator is actually doing when you press the square root button. This guide covers two genuinely different, reliable methods, a manual square root method built on averaging, and a digit by digit square root algorithm borrowed from long division, which grinds out one correct digit at a time with no guessing involved. Every worked example below has been verified digit by digit rather than assumed, so you can follow along and trust the numbers. For an instant answer to check your own hand calculation against, or to explore other root degrees, the root calculator, a general-purpose radical calculator for any root degree, runs the exact same underlying math.
Why bother calculating a square root by hand
Three reasons come up most often. Standardized tests and some classroom exams restrict or forbid calculators for exactly this kind of question, so the manual method is a real, practical skill rather than a curiosity. Building the habit of estimating also sharpens ordinary number sense, the kind that lets you sanity-check whether a calculator result is even in the right ballpark, which matters more than it sounds once decimal points start moving around in a longer calculation, for example catching a misplaced decimal that turns 31.6 into 3.16 before it quietly ruins the rest of a physics or engineering problem. Finally, understanding the manual method demystifies what a calculator or computer is actually doing internally: even modern processors compute square roots using fast variations of the same averaging idea covered below, not by looking up an answer in a table.
A quick mental shortcut is worth knowing even if you never work through a full example: to get a rough one-decimal estimate in your head, find the two nearest perfect squares, see how far your number sits between them as a fraction, and apply that same fraction to the gap between their square roots. The square root of 40 sits four-thirteenths of the way from 36 to 49 (40 is 4 above 36, and the full gap to 49 is 13), so it should land roughly four-thirteenths of the way from 6 to 7, or about 6.3 (the real value is 6.32, close enough for a snap estimate). It is not precise, but it takes seconds and catches an obviously wrong calculator entry instantly. That is also the fastest answer to how to find the square root of a number quickly: bracket it between two perfect squares, then interpolate.
How to calculate a square root by hand, method 1: the averaging (Babylonian) method
This technique, often called the Babylonian method for its earliest known use and closely related to what is now called Newton's method, needs nothing more than a starting guess and repeated division. In parts of South Asia, including Pakistan, the same square root operation this under root formula estimates is sometimes still referred to in everyday speech as finding the "under root" of a number (the number under the radical sign is properly called the radicand), an informal translation that has stuck around from how the radical sign is described in the classroom:
- Make a reasonable starting guess, b, for the square root of your number, a.
- Divide a by b to get c.
- Average b and c: that average is your new, better guess.
- Repeat steps 2 and 3 with the new guess until the answer stops changing at the precision you need.
The reason this works is worth understanding, not just memorizing. Your guess, b, and the result of dividing a by b always land on opposite sides of the true square root, one a little too low and the other a little too high (unless you have already found the exact answer). Averaging the two pulls your next guess toward the middle, between them, and the size of the remaining error shrinks dramatically with each round, roughly squaring smaller every time. That is why even a rough first guess usually reaches three-decimal accuracy in only three or four rounds.
One practical detail matters more than it looks: carry one extra decimal place while you work, beyond how many decimal places you actually want in your final answer, and only round down to your target precision at the very end. Rounding too aggressively at every single step can occasionally make the sequence rock back and forth between two neighboring values instead of settling cleanly, since your calculation can land right on the boundary between two roundings. Carrying that one extra "guard" digit avoids the problem entirely, and it costs almost nothing extra in effort.
This iterative approach exists in the first place because there is no simple square root formula that spits out an exact decimal answer the way the quadratic formula does for a quadratic equation. Most whole numbers are not perfect squares, and most of their square roots are an irrational number, a decimal that runs on forever without ever repeating, so "close enough" to a chosen number of decimal places is the realistic, honest goal, not an exact final digit.
Worked example: the square root of 10
Ten sits between the perfect squares 9 (3 squared) and 16 (4 squared), and much closer to 9, so 3 is a sensible starting guess.
| Step | Divide | Result | New guess (average) |
|---|---|---|---|
| 1 | 10 ÷ 3 | 3.3333 | (3 + 3.3333) ÷ 2 = 3.1667 |
| 2 | 10 ÷ 3.1667 | 3.1579 | (3.1667 + 3.1579) ÷ 2 = 3.1623 |
| 3 | 10 ÷ 3.1623 | 3.1623 | Matches the guess: converged |
The square root of 10 is approximately 3.162. Three short rounds, starting from a plain integer guess, is all it took.
Worked example: the square root of 50
Fifty sits just above the perfect square 49 (7 squared), so 7 is an excellent starting guess, close enough that this example converges even faster than the last one.
| Step | Divide | Result | New guess (average) |
|---|---|---|---|
| 1 | 50 ÷ 7 | 7.1429 | (7 + 7.1429) ÷ 2 = 7.0715 |
| 2 | 50 ÷ 7.0715 | 7.0706 | (7.0715 + 7.0706) ÷ 2 = 7.0711 |
| 3 | 50 ÷ 7.0711 | 7.0710 | (7.0711 + 7.0710) ÷ 2 = 7.0711 |
The square root of 50 is approximately 7.071. Starting close to a perfect square, as this example did, is the single biggest time-saver in this whole method.
Worked example: the square root of 180
For a larger number, 180 sits between 169 (13 squared) and 196 (14 squared), closer to 169, so 13 is the starting guess.
| Step | Divide | Result | New guess (average) |
|---|---|---|---|
| 1 | 180 ÷ 13 | 13.8462 | (13 + 13.8462) ÷ 2 = 13.4231 |
| 2 | 180 ÷ 13.4231 | 13.4097 | (13.4231 + 13.4097) ÷ 2 = 13.4164 |
| 3 | 180 ÷ 13.4164 | 13.4164 | Matches the guess: converged |
The square root of 180 is approximately 13.416. Even starting from a guess that was off by nearly a whole unit, three rounds were still enough.
Choosing a good starting guess
Every example above started from the nearest perfect square, and that is the single most useful habit to build. Keep the first several perfect squares in mind: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. To estimate the square root of any number, find which two perfect squares it falls between, and start your guess at the closer one, nudged slightly toward your actual number. A number sitting almost exactly between two perfect squares, like 40 between 36 and 49, still works fine with the averaging method, it just takes one extra round to fully settle compared to a number sitting right next to a perfect square.
Larger numbers use the same bracketing with bigger squares in mind, and round hundreds and thousands are easy to recognize. The last two rows are not perfect squares, so their answers are rounded.
| Question | Answer | Check |
|---|---|---|
| What is the square root of 1600? | 40 | 40 × 40 = 1600 |
| What is the square root of 2500? | 50 | 50 × 50 = 2500 |
| What is the square root of 10000? | 100 | 100 × 100 = 10000 |
| Square root of 200 | 14.1421 | Between 14² = 196 and 15² = 225 |
| Square root of 1000 | 31.6228 | Between 31² = 961 and 32² = 1024 |
Method 2: the long-division style algorithm
The averaging method needs a guess to start from. This second method does not: it produces one correct digit of the square root at a time, mechanically, similar in spirit to ordinary long division. It takes a bit more setup to learn but never requires estimating anything.
The mechanics, working from the decimal point outward in both directions:
- Pair up the digits of your number starting from the decimal point, working left for the whole-number part and right for the decimal part. 1849 becomes "18" and "49"; 2 (as 2.000000) becomes "02", "00", "00", "00".
- Find the first digit. For the leftmost pair, find the largest digit whose square is less than or equal to that pair. That digit is the first digit of your answer. Subtract its square from the pair to get a remainder.
- Bring down the next pair of digits, attaching it to the remainder.
- Double the answer so far, and find the largest next digit, y, such that (twice the answer so far, with y attached on the end) multiplied by y is less than or equal to the current remainder. That y is the next digit of your answer.
- Subtract and repeat steps 3 and 4, bringing down one new pair of digits each time, for as many digits of accuracy as you need.
Worked example: the square root of 1849 (a perfect square)
A perfect square makes a clean first demonstration, since the remainder reaches exactly zero.
- Pair the digits: "18", "49".
- First pair, 18: the largest digit whose square is at most 18 is 4 (4×4=16). First answer digit: 4. Remainder: 18 − 16 = 2.
- Bring down "49": new remainder is 249. Double the answer so far (4) to get 8. Find the largest digit y such that (80 + y) × y ≤ 249: y = 3 gives 83 × 3 = 249, an exact match. Next answer digit: 3. Remainder: 249 − 249 = 0.
The square root of 1849 is exactly 43 (and 43 × 43 does equal 1849, confirming the result).
Worked example: the square root of 2, to three decimal places
A non-perfect square works exactly the same way, just continuing past the decimal point.
- Pair the digits of 2.000000: "02", "00", "00", "00".
- First pair, 02: the largest digit whose square is at most 2 is 1 (1×1=1). First digit: 1. Remainder: 2 − 1 = 1.
- Bring down "00": remainder becomes 100. Double the answer so far (1) to get 2. Largest y with (20+y) × y ≤ 100: y = 4 gives 24 × 4 = 96. Next digit: 4. Remainder: 100 − 96 = 4.
- Bring down "00": remainder becomes 400. Double the answer so far (14) to get 28. Largest y with (280+y) × y ≤ 400: y = 1 gives 281 × 1 = 281. Next digit: 1. Remainder: 400 − 281 = 119.
- Bring down "00": remainder becomes 11900. Double the answer so far (141) to get 282. Largest y with (2820+y) × y ≤ 11900: y = 4 gives 2824 × 4 = 11296. Next digit: 4. Remainder: 11900 − 11296 = 604.
Reading the digits back with the decimal point placed after the first pair's digit: the square root of 2 is approximately 1.414, matching the well-known value to three decimal places.
Simplifying, adding and multiplying square roots by hand
Sometimes an exact answer beats a decimal. To simplify a square root, factor out the largest perfect square from under the radical sign. The square root of 300 simplified is 10√3, because 300 is 100 × 3. The square root of 60 simplified is 2√15, because 60 is 4 × 15. The square root of 85 simplified is just √85: 85 is 5 × 17, so no square factor comes out. A sqrt simplifier or radical form calculator runs the same factoring, and a simplify square root calculator should agree with this table.
| Radicand | Largest square factor | Simplified | Decimal |
|---|---|---|---|
| 300 | 100 | 10√3 | 17.3205 |
| 200 | 100 | 10√2 | 14.1421 |
| 72 | 36 | 6√2 | 8.4853 |
| 60 | 4 | 2√15 | 7.7460 |
| 85 | none | √85 | 9.2195 |
| 1000 | 100 | 10√10 | 31.6228 |
Once every root is simplified, the operations combine by a few short rules.
| Task | Rule | Example |
|---|---|---|
| How to add square roots | Only roots with the same radicand combine | √12 + √3 = 2√3 + √3 = 3√3 |
| Square root multiplier | √a × √b = √(ab) | √6 × √6 = √36 = 6 |
| How to find the square root of a fraction | Root of the top over root of the bottom | Square root of 1/4 is 1/2, so sqrt 1/4 = 0.5 |
| Square root cubed | The cube of a square root: (√a)³ = a√a | (√2)³ = 2√2, about 2.83 |
The 1/4 square root is a good check on the fraction rule: the root of 1 is 1, the root of 4 is 2, and 1 over 2 is one half.
How to find the square root with a calculator
If you only need the answer, type the number and press the square root key; some models want the key first, then the number. With no such key, raise the number to the power 0.5. A spreadsheet uses =SQRT(A1), and phone calculators show the key once you rotate the phone sideways, which settles how to do square root on calculator apps. The same steps cover how to calculate square root with a calculator on any model. Then compare the result with your hand estimate. A gap bigger than the last digit you kept means one of the two has a slip.
Which method should you actually use
Both are legitimate answers to how to calculate a square root by hand, but for most everyday purposes, the averaging method is faster and easier to hold in your head, especially starting from a nearby perfect square, and it is the method worth practicing first. The long-division method is the more reliable fallback for a number where no good starting guess comes to mind, or in a setting that specifically expects digit-by-digit working, since it never requires a guess at all, only careful arithmetic. Knowing both means you are never stuck: reach for averaging by default, and switch to long division when you would otherwise be guessing blindly.
Common mistakes to avoid
The most common error is rounding too early, at every single step, which can trap the averaging method oscillating between two neighboring answers instead of settling on one, exactly the problem the guard-digit tip above is designed to prevent. A second common mistake in the long-division method is forgetting to pair digits from the decimal point outward correctly, especially adding a leading zero to an odd number of whole-number digits (1849 has 4 digits, an even number, so no padding is needed; a 3-digit number like 169 needs a leading zero, "0169," to pair correctly as "01" and "69"). A third mistake is stopping the averaging method after just one round; the first round from a rough guess is rarely accurate to more than one decimal place, and it is the second and third rounds that do most of the real work.
Where square roots show up outside the classroom
The averaging and long-division methods above are worth practicing on their own, but square roots are not just a classroom exercise. A handful of everyday formulas lean on the same operation, and seeing the numbers work out makes the arithmetic feel less abstract.
The Pythagorean theorem. A right triangle relates its two shorter sides, a and b, to its longest side, c, by a squared plus b squared equals c squared, so finding c from the two shorter sides means taking a square root. A television's advertised diagonal is a direct example: a 16:9 screen with a 55 inch diagonal has 16 squared plus 9 squared equals 337, and the square root of 337 is about 18.358. Dividing 55 by 18.358 gives a scaling factor of about 2.996, so the screen is roughly 16 times 2.996, or 47.9 inches wide, and 9 times 2.996, or 27.0 inches tall.
Free-fall time in physics. Ignoring air resistance, an object falls a height h in the square root of (2h divided by g) seconds. From 20 meters, with g about 9.8 meters per second squared, that is the square root of 4.0816, or about 2.02 seconds. Double the height to 40 meters and the time rises only to about 2.86 seconds, not 4.04, because the height sits inside the square root.
Standard deviation in statistics. Measuring how spread out a data set is involves averaging the squared distance of each value from the mean, called the variance, then taking the square root of that variance to return to the original units, the standard deviation. Squaring first makes every deviation positive before averaging; the square root afterward undoes that squaring so the result is comparable to the data itself. The population versus sample standard deviation guide on this site works through that calculation, square root included, on a real data set.
Recognizing a square root inside a formula, rather than treating the equation as an opaque box, is the number sense this article set out to build.
Check your answer
Both methods above cover how to calculate a square root by hand, and they are worth practicing on numbers you can also verify with a square root calculator. This site's own root calculator computes any square root, cube root, or other whole-number root instantly and also has a worked-through summary of the averaging method for both square roots and higher-degree roots, including a full example of the generalized, weighted-average version used for roots other than square roots.