Area of a square calculator

The area of a square is A = s², the side multiplied by itself. A square is a rectangle whose width and height are equal, so width times height becomes side times side. For a side of 7 the area is 7 × 7 = 49 square units, and the perimeter, four equal sides, is 28.

The word "squared" comes from this shape. Raising a number to the second power is finding the area of the square with that side, and a 7-by-7 grid of unit squares has 49 of them. It is also why area units are called square units: a square foot is a square one foot on each side.

That gives the mistake people make most with squares: "7 square feet" and "7 feet square" are different things. Seven square feet is an area: a patch about 2.65 feet on a side. Seven feet square describes a square that is 7 feet on each side, which has an area of 49 square feet, seven times as much. Listings and floor plans use both phrases, and confusing them gets a carpet or a tile order badly wrong. The other slip is mixing up area and perimeter: 4s = 28 is the distance around, in plain units, and s² = 49 is the space inside, in square units. They are equal only for a square of side 4.

Going backwards from an area to the side takes a square root. A square room of 144 square feet is 12 feet on a side; a square of 49 square meters is 7 meters on a side. If the area is not a perfect square, the side is irrational: a 50 square foot square is √50, about 7.07 feet on a side.

The diagonal is the other number a square is often given by. By the Pythagorean theorem it is s√2, about 9.899 for a side of 7, and the area from the diagonal alone is d² / 2: (7√2)² / 2 = 98 / 2 = 49. That is handy when the corners of a plot or a tile are easier to reach than its edges.

Conversions square the factor. A yard is 3 feet, so a square yard is 9 square feet, and 49 square feet is about 5.444 square yards, not the 16.3 that dividing by 3 would give. A square meter is 10,000 square centimeters for the same reason. Doubling the side quadruples the area: a 14-by-14 square is 196, four times 49.

For a square with unequal sides (a rectangle), use width times height. For a square turned 45 degrees and called a diamond, it is still s², or half the product of the diagonals, which is the rhombus rule.