Area is the quantity that expresses the extent of a two-dimensional figure or shape or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. 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 or the volume of a solid.
The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. 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. 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.
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 an 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. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.
Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.
An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of special kind of plane figures to the set of real numbers, which satisfies the following properties:
  • For all S in M, a ≥ 0.
  • If S and T are in M then so are ST and ST, and also a = a + aa.
  • If S and T are in M with ST then TS is in M and a = aa.
  • If a set S is in M and S is congruent to T then T is also in M and a = a.
  • Every rectangle R is in M. If the rectangle has length h and breadth k then a = hk.
  • Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. SQT. If there is a unique number c such that a ≤ c ≤ a for all such step regions S and T, then a = c.
It can be proved that such an area function actually exists.
made of PVC pipe.
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 measured in square metres, square centimetres, square millimetres, square kilometres, square feet, square yards, square miles, 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.

Calculation of the area of a square whose length and width are 1 metre would be:
1 metre × 1 metre = 1 m2
and so, a rectangle with different sides would have an area in square units that can be calculated as:
3 metres × 2 metres = 6 m2. This is equivalent to 6 million square millimetres. Other useful conversions are:
In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units.
the relationship between square feet and square inches is
where 144 = 122 = 12 × 12. Similarly:
In addition, conversion factors include: