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Area is a quantity that expresses the extent of a two-dimensional surface or shape, or planar lamina, 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 (written as m2), 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 (termed measurable sets) to the set of real numbers which satisfies the following properties:
It can be proved that such an area function actually exists.
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 (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 conversion between two square units is the square of the conversion between the corresponding length units. For example, since
the relationship between square feet and square inches is
where 144 = 122 = 12 × 12. Similarly:
The acre is also commonly used to measure land areas, where
An acre is approximately 40% of a hectare.
The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom. On the other hand, if geometry is developed before arithmetic, this formula can be used to define multiplication of real numbers.
For an example, any parallelogram can be subdivided into a trapezoid and a right triangle, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle:
However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of the parallelogram:
The formula for the area of a circle (more properly called area of a disk) is based on a similar method. Given a circle of radius r, it is possible to partition the circle into sectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form and approximate parallelogram. The height of this parallelogram is r, and the width is half the circumference of the circle, or πr. Thus, the total area of the circle is r × πr, or πr2:
Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of the areas of the approximate parallelograms is exactly πr2, which is the area of the circle.
This argument is actually a simple application of the ideas of calculus. In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus. Using modern methods, the area of a circle can be computed using a definite integral:
Most basic formulae for surface area can be obtained by cutting surfaces and flattening them out. For example, if the side surface of a cylinder (or any prism) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone, the side surface can be flattened out into a sector of a circle, and the resulting area computed.
The formula for the surface area of a sphere is more difficult to derive: because the surface of a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by Archimedes in his work On the Sphere and Cylinder. The formula is:
where r is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus.
(see Green's theorem) or the z-component of
The general formula for the surface area of the graph of a continuously differentiable function where and is a region in the xy-plane with the smooth boundary:
There are formulae for many different regular and irregular polygons, and those additional to the ones above are listed here.
|Regular triangle (equilateral triangle)||is the length of one side of the triangle.|
|Triangle||is half the perimeter, , and are the length of each side.|
|Triangle||and are any two sides, and is the angle between them.|
|Triangle||and are the base and altitude (measured perpendicular to the base), respectively.|
|Rhombus||and are the lengths of the two diagonals of the rhombus.|
|Parallelogram||is the length of the base and is the perpendicular height.|
|Trapezoid||and are the parallel sides and the distance (height) between the parallels.|
|Regular hexagon||is the length of one side of the hexagon.|
|Regular octagon||is the length of one side of the octagon.|
|Regular polygon||is the side length and is the number of sides.|
|Regular polygon||is the perimeter and is the number of sides.|
|Regular polygon||is the radius of a circumscribed circle, is the radius of an inscribed circle, and is the number of sides.|
|Regular polygon||is the apothem, or the radius of an inscribed circle in the polygon, and is the perimeter of the polygon.|
|Circle||is the radius and the diameter.|
|Circular sector||and are the radius and angle (in radians), respectively and is the length of the perimeter.|
|Ellipse||and are the semi-major and semi-minor axes, respectively.|
|Total surface area of a cylinder||and are the radius and height, respectively.|
|Lateral surface area of a cylinder||and are the radius and height, respectively.|
|Total surface area of a sphere||and are the radius and diameter, respectively.|
|Total surface area of a pyramid||is the base area, is the base perimeter and is the slant height.|
|Total surface area of a pyramid frustum||is the base area, is the base perimeter and is the slant height.|
|Square to circular area conversion||is the area of the square in square units.|
|Circular to square area conversion||is the area of the circle in circular units.|
The above calculations show how to find the area of many common shapes.
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