Squaring a number — multiplying it by itself — is the engine behind every area calculation on earth. A 12-foot by 12-foot bedroom is 12² = 144 square feet of flooring to buy; a pizza's price-per-square-inch argument rests on radius²; and the x² term dominates every quadratic equation. The square grows fast: 10² is 100, 20² is 400, and that quadratic growth is why doubling a room's side quadruples its area.

This tool computes the three powers people reach for most. Type any value in the Number (x) box and read the x² (Square) output, the √x (Square Root) output, and the x³ (Cube) output together. Entering 12 shows 144, about 3.464, and 1728 in one glance — the area of the bedroom floor, the side of a square with area 12, and the volume of a 12-inch cube, all from a single input.

Perfect squares, memorized.

The fastest way to get fluent with squares is to memorize the squares of the whole numbers from 1 through 20, because they come up everywhere. One squared is 1, 2 squared is 4, 3 squared is 9, 4 squared is 16, 5 squared is 25, 6 squared is 36, 7 squared is 49, 8 squared is 64, 9 squared is 81, 10 squared is 100, 11 squared is 121, 12 squared is 144, 13 squared is 169, 14 squared is 196, 15 squared is 225, 16 squared is 256, 17 squared is 289, 18 squared is 324, 19 squared is 361, and 20 squared is 400. Notice the pattern in the endings: squares of numbers ending in 5 always end in 25, and squares of even numbers are always even. A 13-foot square patio is 169 square feet, a 17-inch square tile field is 289 square inches, and a 20-foot square garden bed is 400 square feet. Once these are automatic, you can estimate any square by bracketing it: 14 squared is 196, so 14.5 squared must be a little above 196, and indeed it is 210.25. That bracketing habit is what makes a calculator a check rather than a crutch.

Negatives, decimals, and fractions.

Squaring erases the sign of the input, but only when the parentheses say so, and this is where careful students earn their marks. Negative 5 squared, written with parentheses as (-5) squared, is 25, because negative 5 times negative 5 is positive 25. But a minus sign written outside the square, as in -(5 squared), means square first and then negate, giving negative 25. The two expressions look almost identical and differ by 50, so always check where the minus sign sits. Decimals square the same way, with one extra thing to watch: the decimal point moves twice as far. One half squared is 0.25, because 0.5 times 0.5 is 0.25, and 2.5 squared is 6.25. Fractions behave identically: one half squared is one quarter, two thirds squared is four ninths, and three quarters squared is nine sixteenths. A useful check is that squaring a number between 0 and 1 makes it smaller, while squaring a number above 1 makes it bigger. If your answer for 0.8 squared comes out above 0.8, you have gone wrong somewhere, because 0.8 squared is 0.64.

Square, double, and cube are three different operations.

Three operations get confused more than any others in algebra, and a side-by-side look ends the confusion. Doubling 12 gives 24, squaring 12 gives 144, and cubing 12 gives 1728. Doubling adds the number to itself once, squaring multiplies it by itself once, and cubing multiplies it by itself twice. The gaps grow fast: for 7, double is 14, square is 49, cube is 343; for 20, double is 40, square is 400, cube is 8000. This is why the phrase "twice as big" is so dangerous in real life. A room twice as wide is not twice the area, it is four times the area, and a box twice as long on every side is not twice the volume, it is eight times the volume. When a contractor says a floor is "double the size," ask whether they mean double the side or double the area, because the price difference between those two readings is a factor of two. Keep the three operations in separate mental boxes and label every answer with its units: feet for a length, square feet for an area, cubic feet for a volume.

Area, scaling, and the price of flooring.

Area is where squaring earns its keep in daily life. A square room 12 feet on a side needs 144 square feet of flooring, and at 6 dollars a square foot that is 864 dollars of material before waste and cuts. A 15-foot square room needs 225 square feet, which at 4 dollars a square foot is 900 dollars. The scaling rule is the thing to internalize: multiply the side by 2 and the area multiplies by 4; multiply the side by 3 and the area multiplies by 9; halve the side and the area quarters. A 10-foot square patio is 100 square feet, but a 20-foot square patio is 400 square feet, not 200. That is why doubling a garden bed's side does not double the soil or the seed: both quadruple. The same rule runs in reverse for shrinking: a rug that is half the side of another covers one quarter of the floor. Tile setters use this constantly, because a room that looks only a little bigger can need dramatically more tile, and the waste factor, usually 10 percent for straight runs and more for diagonal patterns, applies to the squared area, not to the side length.

Squares inside bigger formulas.

Squaring rarely appears alone; it is usually one step inside a larger formula, and recognizing it there saves a lot of confusion. The area of a circle is pi times the radius squared, so a circle with a 5-foot radius covers about 78.54 square feet, and one with a 10-foot radius covers about 314.16 square feet, four times as much, because the radius doubled. The Pythagorean theorem is a sum of two squares: a right triangle with legs 3 and 4 has a hypotenuse of 5, because 9 plus 16 is 25, and the square root of 25 is 5. Legs 5 and 12 give a hypotenuse of 13, because 25 plus 144 is 169. The diagonal of a square room 12 feet on a side is the square root of 144 plus 144, which is the square root of 288, about 16.97 feet, the length of the longest straight rug or cable run corner to corner. In physics, kinetic energy depends on velocity squared, which is why doubling a car's speed quadruples its stopping energy, and in statistics, variance is the average of squared distances from the mean. Whenever you see a squared term, ask what quantity is being multiplied by itself and why: the answer is usually an area, a distance squared, or an energy.

Mental-math tricks that actually work.

A few squaring shortcuts are reliable enough to use with confidence. Any number ending in 5 squares by a clean rule: take the tens digit, multiply it by one more than itself, and append 25. For 25, 2 times 3 is 6, so 25 squared is 625. For 35, 3 times 4 is 12, so 35 squared is 1225. For 15, 1 times 2 is 2, so 15 squared is 225. Numbers near 100 square through the identity (100 minus a) squared equals 10000 minus 200a plus a squared: 98 squared is 10000 minus 400 plus 4, which is 9604. The general expansion (a plus b) squared equals a squared plus twice a times b plus b squared, which turns 103 squared into 10000 plus 600 plus 9, or 10609, with no long multiplication at all. These tricks are not party games; they are error checks. If the calculator says 25 squared is 6250, the ends-in-25 rule tells you instantly that a decimal point slipped. Use the calculator for the final answer and a trick for the sanity check, and the two will catch each other's mistakes.

How to use the calculator above.

Type any number in the Number box and read three outputs at once. Enter 12 and you get 144 for the square, about 3.46410163 for the square root, and 1728 for the cube. Enter 7, the tool's starting value, and you get 49, about 2.64575131, and 343. Enter a negative number and the square output is still positive, because a negative times a negative is positive, while the square-root output is the principal root of the number entered. Enter a decimal like 2.5 and you get 6.25, 1.58113883, and 15.625 in one glance. The three outputs together answer three different questions: how big is the area, what side gives this area, and how big is the volume. For a flooring estimate, the square output is the one that matters; for reversing an area back to a side length, the square-root output does the work; for a cube-shaped container, the cube output gives the volume directly. Try 0.5 for a tidy illustration: 0.25, about 0.70710678, and 0.125, three answers that show the square shrinking below the input, exactly as the rule for numbers under 1 predicts.

Mistakes to avoid.

The classic error is squaring a sum as if it distributed: (3 plus 4) squared is 49, not 9 plus 16, which is 25. Squares do not split across addition; add first, then square. The second error is the sign trap above: (-5) squared is 25, but -(5 squared) is negative 25, and the parentheses decide. The third is unit forgetfulness: squaring 12 feet gives 144 square feet, not 144 feet, and a price quoted per square foot must multiply the squared number, not the side. The fourth is using the diameter where a formula wants the radius: a circle with a 10-foot diameter has a 5-foot radius, and its area is about 78.54 square feet, not the 314.16 you get by squaring the diameter. The fifth is rounding mid-problem: 3.464 rounded to 3.46, squared, gives 11.9716, not 12, and that drift compounds across a project. Keep full precision until the last step, write the units on every answer, and square the answer back as a check whenever the stakes are a purchase or a cut.

With the pre-filled defaults — Number (x) 7 — the Square Calculator returns x² (Square) 49.00000000, √x (Square Root) 2.64575131, x³ (Cube) 343.00000000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.