Area | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-25

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Area


00:02:23 1 Formal definition
00:03:40 2 Units
00:03:47 2.1 Conversions
00:04:32 2.1.1 Non-metric units
00:05:12 2.2 Other units including historical
00:05:27 3 History
00:05:31 3.1 Circle area
00:05:57 3.2 Triangle area
00:06:20 3.3 Quadrilateral area
00:07:02 3.4 General polygon area
00:07:13 3.5 Areas determined using calculus
00:07:40 4 Area formulas
00:08:06 4.1 Polygon formulas
00:08:50 4.1.1 Rectangles
00:08:53 4.1.2 Dissection, parallelograms, and triangles
00:09:02 4.2 Area of curved shapes
00:10:59 4.2.1 Circles
00:12:02 4.2.2 Ellipses
00:12:36 4.2.3 Surface area
00:12:59 4.3 General formulas
00:13:24 4.3.1 Areas of 2-dimensional figures
00:13:33 4.3.2 Area in calculus
00:14:56 4.3.3 Bounded area between two quadratic functions
00:15:15 4.3.4 Surface area of 3-dimensional figures
00:15:30 4.3.5 General formula for surface area
00:15:49 4.4 List of formulas
00:17:06 4.5 Relation of area to perimeter
00:17:16 4.6 Fractals
00:19:06 5 Area bisectors
00:19:35 6 Optimization
00:20:24 7 See also
00:20:37 8 References
00:20:47 9 External links
00:24:30 Area in calculus
00:28:29 Bounded area between two quadratic functions
00:30:40 Surface area of 3-dimensional figures
00:33:39 General formula for surface area
00:36:19 List of formulas
00:36:39 Relation of area to perimeter
00:37:58 Fractals
00:38:32 Area bisectors
00:39:26 Optimization
00:40:55 See also



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SUMMARY
=======
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 (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.