Area of a Square Calculator

What the area of a square measures

The area of a square is the amount of flat surface it covers. A square has four equal sides that meet at right angles, so a single measurement, the side length, fixes the whole shape and therefore its area.

Area is counted in square units. A square whose side is 1 centimetre covers one square centimetre (cm²); a square whose side is 1 mile covers one square mile (mile²). This calculator keeps the unit you pick for the side and reports the answer in the matching squared unit, so a side in feet gives an area in square feet.

Area of a square formula

Multiply the side by itself. That is the reason the operation is called squaring a number.

A = s × s = s²A is the area and s is the length of one side.

You do not always start with the side. Two other measurements lead to the same area:

• From the diagonal d. The diagonal cuts the square into two right triangles whose legs are the sides, so d² = 2s². Therefore A = d² ÷ 2.
• From the perimeter P. A square has four equal sides, so s = P ÷ 4 and A = (P ÷ 4)².
• Back to the side. If only the area is known, the side is its square root. The square root calculator does that step for you.

How to use the calculator

1. Enter the length of one side in the Square Side box. The field starts at 3 so you can see an example straight away.
2. Choose the unit next to it: millimetres, centimetres, decimetres, metres, kilometres, inches, feet, yards or miles.
3. Press Calculate area. The result appears under the form with the unit squared, for example 56.25 m².

Whole-number areas are shown without a decimal point. Other results are rounded to at most six decimal places, and very large areas are printed digit by digit rather than in scientific notation. The side must be zero or larger; a negative or empty entry is rejected.

Worked examples

Example 1: a patio. A square patio has sides of 7.5 m. Area = 7.5 × 7.5 = 56.25 m². Any quantity that is priced per square metre, such as paving slabs or sealant, can be estimated from that figure.

Example 2: flooring in yards. A square room measures 12 ft along each wall, so its area is 12 × 12 = 144 ft². One square yard is 3 ft × 3 ft = 9 ft², hence the room is 144 ÷ 9 = 16 yd².

Example 3: from the diagonal. The diagonal of a square plate is 10 cm. Then A = 10² ÷ 2 = 50 cm², and each side is √50 ≈ 7.0711 cm.

Example 4: from the perimeter. A fence around a square plot is 36 m long. Each side is 36 ÷ 4 = 9 m, so the plot covers 9² = 81 m².

Scaling and unit conversion

Area grows with the square of the side. Doubling the side makes the area four times larger, tripling it makes the area nine times larger, and halving it leaves a quarter. This is why a 20 cm square tile covers four times as much floor as a 10 cm one, not twice as much.

The same square can be described in different units, and the area converts with the square of the length factor:

• 1 m² = 100 cm × 100 cm = 10,000 cm²
• 1 yd² = 3 ft × 3 ft = 9 ft²
• 1 ft² = 144 in²
• 1 m² ≈ 10.7639 ft²

To switch an area between units after the fact, use the area unit converter. To convert the side before you start, the length converter is the quicker route.

Common mistakes

• Mixing up area and perimeter. Perimeter is a length (4 × s) and is measured in plain units; area is a surface and is measured in square units. The perimeter of a square calculator covers the first.
• Forgetting to square the unit. 9 cm² is not 9 cm. Writing the unit correctly is part of the answer.
• Treating the diagonal as a side. A diagonal is longer than a side by a factor of about 1.4142 (the square root of 2). Use the diagonal formula above if that is what you measured.
• Converting the area by the length factor. Going from metres to centimetres multiplies lengths by 100 but areas by 100 × 100 = 10,000.

Related shapes

A square is a rectangle with equal sides, so the area of a rectangle calculator gives the same answer when width and length match. Other flat shapes follow similar one-line rules: the triangle, the trapezoid, the parallelogram and the circle. For a regular six-sided figure, try the hexagon calculator.