# Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (m2), which is the area of a square whose sides are one metre long[1]. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles[2]. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus[3].

For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.

Area plays a important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.[4] In analysis, the area of a subset of the plane is defined using Lebesgue measure[5], though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.

## Units

Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measure in square metres (m2), square centimetres (cm2), square millimetres (mm2), square kilometres (km2), square feet (ft2), square yards (yd2), square miles (mi2), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units.

The SI unit of area is the square metre, which is considered an SI derived unit.

### Conversions

The conversion between two square units is the square of the conversion between the corresponding length units. For example, since

1 foot = 12 inches,

the relationship between square feet and square inches is

1 square foot = 144 square inches,

where 144 = 122 = 12 × 12. Similarly:

• 1 square kilometer = 1,000,000 square meters
• 1 square meter = 10,000 square centimetres = 1,000,000 square millimetres
• 1 square centimetre = 100 square millimetres
• 1 square yard = 9 square feet
• 1 square mile = 3,097,600 square yards = 27,878,400 square feet

In addition,

• 1 square inch = 6.4516 square centimetres
• 1 square foot = square metres
• 1 square yard = square metres
• 1 square mile = square kilometres

### Other units

There are several other common units for area. The are was the original unit of area in the metric system, with

• 1 are = 100 square metres

Though the are has fallen out of use, the hectare is still commonly used to measure land:

• 1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres

Other uncommon metric units of area include the tetrad, the hectad, and the myriad.

The acre is also commonly used to measure land areas, where

• 1 acre = 4,840 square yards = 43,560 square feet.

An acre is approximately 40% of a hectare.