A square backyard covering 900 square feet has sides of exactly 30 feet, because the square root undoes squaring. The square root of a number is the value that, multiplied by itself, returns the original: the square root of 900 is 30, since 30 times 30 is 900. Gardeners converting area to fence length, tilers figuring how many 12-inch squares span a room, and photographers computing f-stops all lean on roots. The symbol is really a question: what times itself gives this number?

Perfect squares first.

The easiest roots are the ones you already know by heart, because they are the squares of whole numbers. The square root of 1 is 1, of 4 is 2, of 9 is 3, of 16 is 4, of 25 is 5, of 36 is 6, of 49 is 7, of 64 is 8, of 81 is 9, of 100 is 10, of 121 is 11, and of 144 is 12. Learn the squares through 15 squared, which is 225, and through 20 squared, which is 400, and you will recognize a perfect square on sight instead of grinding through division. A 144-square-foot patio is 12 feet on a side, a 2,500-square-foot lot that happens to be square is 50 feet on a side, and a 10,000-square-foot plot is 100 feet on a side. Every one of those answers is exact, with no decimal tail at all, because the input was a perfect square.

The principal root and its negative twin.

Every positive number actually has two square roots, one positive and one negative, because a negative times a negative is positive. The square root of 900 is 30, but negative 30 times negative 30 is also 900. The radical symbol denotes the principal root, the positive one, by convention: when a problem asks for the square root of 900, the answer is 30, not plus-or-minus 30. The plus-or-minus shows up when you solve an equation: x squared equals 900 has two solutions, x equals 30 and x equals negative 30, and dropping the negative one loses half the answer. Context decides which you need. A fence length is never negative 30 feet, so the principal root is the whole story; a quadratic equation keeps both. Zero is its own special case: the square root of 0 is 0, the only number that is its own negative twin. Negative inputs have no real square root at all: the square root of negative 9 is 3i, an imaginary number, which is why this calculator expects zero or positive inputs.

Irrational roots and why the decimals never end.

Most numbers are not perfect squares, and their roots are irrational: decimals that never repeat and never terminate. The square root of 2 is about 1.414214, and it is the diagonal of a 1-by-1 square, a fact the ancient Greeks found so disturbing that legend says they kept it quiet. The square root of 3 is about 1.732051, of 5 is about 2.236068, of 10 is about 3.162278, of 20 is about 4.472136, and of 50 is about 7.071068. None of these can be written as an exact fraction, so every decimal you see is rounded. That is why this calculator gives you two outputs: the full square root and a rounded-to-6-decimals version. Six decimals is plenty for building, sewing, and cooking, but if you are chaining a root into further calculations, carry the unrounded value as far as you can and round only at the end. Rounding early is how a 1,000-foot fence run comes out a foot short.

Simplifying radicals instead of reaching for decimals.

Before decimals, mathematicians simplify: pull the perfect squares out of the radical. The square root of 72 is the square root of 36 times 2, which is 6 times the square root of 2, or about 8.485281. The square root of 50 is 5 times the square root of 2, about 7.071068. The square root of 200 is 10 times the square root of 2, about 14.142136. The square root of 12 is 2 times the square root of 3, about 3.464102. The square root of 48 is 4 times the square root of 3, about 6.928203. The square root of 300 is 10 times the square root of 3, about 17.320508. Simplified form is exact where a decimal is approximate, and it is what teachers want on a test: 6 times the square root of 2 is the answer, 8.49 is the approximation. To simplify any root, factor the number, pair up identical factors, and pull one from each pair outside: 144 factors as 2 times 2 times 2 times 2 times 3 times 3, the pairs give 2 times 2 times 3, which is 12.

Finding a root without a calculator.

Three methods cover nearly every by-hand situation. First, estimation with averaging, the Babylonian method: to find the square root of 20, guess 10, divide 20 by 10 to get 2, average 10 and 2 to get 6, divide 20 by 6 to get 3.3333, average 6 and 3.3333 to get 4.6667, divide 20 by 4.6667 to get 4.2857, average to get 4.4762, and one more round lands at 4.4719, essentially the true 4.472136. Each round roughly doubles the correct digits, so three or four rounds beat any ruler. Second, prime factorization for perfect squares: factor 144 into 2 to the fourth times 3 squared, take one factor from each pair, and multiply 2 times 2 times 3 to get 12. Third, the long-division method, the digit-by-digit algorithm taught before calculators: group the digits in pairs from the decimal point, find the largest square that fits the first pair, subtract, double the root so far, and find the next digit. For 3249, the pairs are 32 and 49, the largest square under 32 is 25, and the method finishes at 57, since 57 times 57 is 3249. A textbook example worth memorizing is the square root of 152.2756, which is exactly 12.34, because 12.34 times 12.34 is 152.2756.

Where roots actually show up.

The Pythagorean theorem is the most common: a right triangle with legs 3 and 4 has a hypotenuse of the square root of 9 plus 16, which is the square root of 25, which is 5. Legs 5 and 12 give a hypotenuse of 13, since 25 plus 144 is 169. A rectangular garden 30 feet by 40 feet has a diagonal of the square root of 900 plus 1600, which is the square root of 2500, which is 50 feet: the string you stretch corner to corner. Area-to-side conversions run the other direction: a square room of 144 square feet is 12 feet on a side, and a square tile field of 900 square feet needs 30 feet of edging per side. Statistics uses roots constantly: if a data set has a variance of 16, its standard deviation is the square root of 16, which is 4, the typical distance from the average. Physics uses them for speed from energy and distance from time. And every phone screen diagonal, every camera f-stop sequence, and every map scale that converts area to length hides a square root.

How to use the calculator above.

Type the number in the Number box and read three outputs. The Square Root output is the principal root: enter 2 and you get about 1.414214, enter 900 and you get 30. The Rounded output gives the same value tidied to 6 decimal places, which is what you write on a cut list or a homework line. The verification output multiplies the root by itself and should return your original number: for 2 it returns 2, closing the loop and proving the answer. Try 144 for a clean 12, try 10 for 3.162278, try 0.25 for 0.5, and try 1,000,000 for a satisfying 1000. If you enter a negative number, remember the tool is built for real roots: the answer lives in imaginary numbers, not on this page.

Mistakes to avoid.

The classic error is distributing a root over a sum: the square root of 9 plus 16 is 5, not 3 plus 4, which is 7. Roots do not split across addition. Another is forgetting the negative root when solving equations: x squared equals 144 means x is 12 or negative 12. A third is squaring the wrong thing when checking: to verify the square root of 20 is about 4.4721, square 4.4721, which gives about 20.0000, and do not confuse the check with the root itself. A fourth is rounding mid-problem: the square root of 10 rounded to 3.16, squared, gives 9.9856, not 10, and that drift compounds across a project. And a fifth is reading the symbol as asking for half: the square root of 100 is 10, not 50, because 10 times 10 is 100 while 50 times 2 is a different operation entirely. Keep the definition in front of you, what times itself gives this, and every one of these traps disappears.

One last habit worth building: whenever a root looks suspicious, square it. The square root of 152.2756 looks like a strange number until 12.34 times 12.34 returns 152.2756 exactly. That round trip is the whole discipline of roots in one move: guess, square, adjust, and trust only the answer that multiplies back to where you started.

With the pre-filled defaults — Number (x) 144 — the Square Root Calculator returns Square Root (√x) 12.0000000000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.