NationMaster - Encyclopedia: Area (geometry)

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, although there are space-filling curves. 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 kind of property which exists as magnitude or multitude. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... 2-dimensional renderings (ie. ... An open surface with X-, Y-, and Z-contours shown. ... Space-filling curves or Peano curves are curves, first described by Giuseppe Peano, whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube). ... For other uses, see Volume (disambiguation). ...

Bold text==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. MOOSI! 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 in Wiktionary, the free dictionary. ...

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

The area of the unit square is equal to one.
Congruent polygons have equal areas.
(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. The unit square in a Cartesian coordinate system with coordinates (x,y) is defined as the square consisting of the points where both x and y lie in the unit interval from 0 to 1. ... An example of congruence. ... Look up Additive in Wiktionary, the free dictionary. ...

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 (2a-45 is stupid) 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 (Geometry makes no sense)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 (all over 2a)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 length, area or 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. ... A ball can be decomposed into a finite number of pieces and reassembled into two balls identical to 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. For other uses, see Volume (disambiguation). ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... 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. ... Look up content in Wiktionary, the free dictionary. ...

Formulas

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

Image File history File links Areabetweentwographs. ... Image File history File links Areabetweentwographs. ...

Areas of 2-dimensional figures

square: $s^2$ (where s is the length of one side).
rhombus (includes "kites"): $frac{1}{2}Dd$ (where D is the length of one diagonal and d is the length of the other diagonal)
circle: $π r 2$ (where r is the radius)
ellipse: $π a b$ (where a and b are the semi-major and semi-minor axes)
any regular polygon: $frac{Pa}{2}$ (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: $B h$ (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: $frac{(B + b)h}{2}$ (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: $sqrt{s(s-a)(s-b)(s-c)}$ (where a, b, c are the sides of the triangle, and $s = frac{a + b + c}{2}$ is half of its perimeter) If an angle and its two included sides are given, then area=absinC where C is the given angle and a and b are its included sides. If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of (x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂) all divided by 2. This formula is also known as the shoelace formula and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points, (x₁,y₁) (x₂,y₂) (x₃,y ₃) Another approach for a coordinate triangle is to use calculus to find the area.
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 ${1 over 2} int_0^{2pi} r^2 , dtheta$ .
the area enclosed by a parametric curve $vec u(t) = (x(t), y(t))$ with endpoints $vec u(t_0) = vec u(t_1)$ is given by the line integrals

$oint_{t_0}^{t_1} x dot y , dt = - oint_{t_0}^{t_1} y dot x , dt = {1 over 2} oint_{t_0}^{t_1} (x dot y - y dot x) , dt$

(see Green's theorem) Circle illustration This article is about the shape and mathematical concept of circle. ... For other uses, see Ellipse (disambiguation). ... The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... A parallelogram. ... This article is about the geometric figure. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A triangle with sides a, b, and c. ... For other uses, see Calculus (disambiguation). ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... This article is about the concept of integrals in calculus. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ... 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. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... 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 two-dimensional case of...

or the z-component of

${1 over 2} oint_{t_0}^{t_1} vec u times dot{vec u} , dt$

all over 2a all over 2a all over 2a all over 2a Que la gay?

Surface area of 3-dimensional figures

cube: $6 s 2$ , where s is the length of any side
rectangular box: $2 (ell w + ell h + w h)$ , where l, w, and h are the length, width, and height of the box
sphere: $4π r 2$ , 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: $2π r (h + r)$ , where r is the radius of the circular base, and h is the height
cone: $pi r (r + sqrt{r^2 + h^2})$ , where r is the radius of the circular base, and h is the height. That can also be rewritten as $π r 2 + π r l$ where r is the radius and l is the slant height of the cone. $π r 2$ is the base area while $π r l$ is the lateral surface area of the cone.
prism: 2 * Area of Base + Perimeter of Base * Height

The general formula for the surface area of the graph of a continuously differentiable function $z = f (x, y),$ where $(x,y)in Dsubsetmathbb{R}^2$ and $D$ is a region in the xy-plane with the smooth boundary: Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... In anatomy, the cuboid bone is a bone in the foot. ... For other uses, see Sphere (disambiguation). ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... This article is about the geometric object, for other uses see Cone. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ...

$A=iint_Dsqrt{left(frac{partial f}{partial x}right)^2+left(frac{partial f}{partial y}right)^2+1}; dx dy.$

Even more general formula for the area of the graph of a parametric surface in the vector form $mathbf{r}=mathbf{r}(u,v),$ where $mathbf{r}$ is a continuously differentiable vector function of $(u,v)in Dsubsetmathbb{R}^2$ : A parametric surface is a surface defined by a parametric equation, involving two parameters, most commonly (s, t) or (u,v). ...

$A=iint_D left|frac{partialmathbf{r}}{partial u}timesfrac{partialmathbf{r}}{partial v}right|;du dv.$

Categories: Euclidean plane geometry | Area

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