Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. Points and lines have zero area. Depending on the particular definition taken, a figure may have infinite area, for example the entire Euclidean plane. In three dimensions, the analog of area is called a volume. Quantity is a generic term used when referring to the measurement (count, amount) of a scalar, vector, number of items or to some other way of denominating the value of a collection or group of items. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... Jump to: navigation, search Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... An open surface with X-, Y-, and Z-contours shown. ... Jump to: navigation, search Volume, also called capacity, is a quantification of how much space an object occupies. ...

How to define area

Although area seems to be one of the basic notions in geometry, it is not at all easy to define even in the Euclidean plane. Most textbooks avoid defining an area, relying on self-evidence. To make the concept of area meaningful one has to define it, at the very least, on polygons in the Euclidean plane, and it can be done using the following definition: In epistemology, a self-evident proposition is one that can be understood only by one who knows that it is true. ... Look up Polygon on Wiktionary, the free dictionary For other use please see Polygon (disambiguation) A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ...

The area of a polygon in the Euclidean plane is a positive number such that:

1. The area of the unit square is equal to one.
2. Congruent polygons have equal areas.
3. (additivity) If a polygon is a union of two polygons which do not have common interior points, then its area is the sum of the areas of these polygons.

But before using this definition one has to prove that such an area indeed exists. In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ...

In other words, one can also give a formula for the area of an arbitrary triangle, and then define the area of an arbitrary polygon using the idea that the area of a union of polygons (without common interior points) is the sum of the areas of its pieces. But then it is not easy to show that such area does not depend on the way you break the polygon into pieces.

Nowadays, the most standard (correct) way to introduce area is through the more advanced notion of Lebesgue measure, but one should note that in general, if one adopts the axiom of choice then it is possible to prove that there are some shapes whose Lebesgue measure cannot be meaningfully defined. Such 'shapes' (they cannot a fortiori be simply visualised) enter into Tarski's circle-squaring problem (and, moving to three dimensions, in the Banach-Tarski paradox). The sets involved do not arise in practical matters. In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... Tarskis circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a circle (including its interior) in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. ... The Banach-Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original. ...

In three dimensions, the analog of area is called volume. The n dimensional analog is defined by means of a measure or as a Lebesgue integral; this is sometimes referred to as content. Jump to: navigation, search Volume, also called capacity, is a quantification of how much space an object occupies. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... Content can mean Comfort and a feeling of satisfaction Creations, as in open content or free content. ...

Formulae

The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions

area between two graphs To compute area between two curves , say f(x) and g(x), for a<x<b, We may first compute area under the curve A1 = f(x) and area under the curve A2 = g(x). ... area between two graphs To compute area between two curves , say f(x) and g(x), for a<x<b, We may first compute area under the curve A1 = f(x) and area under the curve A2 = g(x). ...

Areas of 2-dimensional figures

• square or rectangle: (where l is the length and w is the width; in the case of a square, l = w.
• circle: πr² (where r is the radius)
• ellipse: πab (where a and b are the semi-major and semi-minor axes)
• any regular polygon: (where P = the length of the perimeter, and a is the length of the apothem of the polygon [the distance from the center of the polygon to the center of one side])
• a parallelogram: Bh (where the base B is any side, and the height h is the distance between the lines that the sides of length B lie on)
• a trapezoid: (B and b are the lengths of the parallel sides, and h is the distance between the lines on which the parallel sides lie)
• a triangle: (where B is any side, and h is the distance from the line on which B lies to the other vertex of the triangle). This formula can be used if the height h is known. If the lengths of the three sides are known then Heron's formula can be used: (where a, b, c are the sides of the triangle, and is half of its perimeter)
• the area between the graphs of two functions is equal to the integral of one function, f(x), minus the integral of the other function, g(x).
• an area bounded by a function r = r(θ) expressed in polar coordinates is .
• the area enclosed by a parametric curve with endpoints is given by the path integrals

(see Green's theorem) Jump to: navigation, search In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). ... In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. ... In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas. ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ... A parallelogram. ... A trapezoid (American English) or trapezium (British English) is a quadrilateral two of whose sides are parallel to each other. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In geometry, Herons formula (also called Heros formula) states that the area of a triangle whose sides have lengths a, b and c is where s is the triangles semiperimeter: (see also square root). ... See also the disambiguation page title equality. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, subtraction is one of the four basic arithmetic operations. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... In mathematics, a path integral (also known as a line integral) is an integral where the function to be integrated is evaluated along a path or curve. ... In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special case of the more...

or the z-component of

Surface area of 3-dimensional figures

• cube: 6s², where s is the length of any side
• rectangular box: , where l, w, and h are the length, width, and height of the box
• sphere: 4πr², where π is the ratio of circumference to diameter of a circle, 3.14159..., and r is the radius of the sphere
• ellipsoid: see the article
• cylinder: 2πr(h + r), where r is the radius of the circular base, and h is the height
• cone: , where r is the radius of the circular base, and h is the height.