Area (geometry)

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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.

1 How to define area
2 Formulas
- 2.1 Areas of 2-dimensional figures
- 2.2 Surface area of 3-dimensional figures

[edit] 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:

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.

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 small 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 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.

[edit] Formulas

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

[edit] 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: $\frac{Bh}{2}$ (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 ${1 \over 2} \int_0^{2\pi} r^2 \, d\theta$ .
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)

or the z-component of

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

[edit] Surface area of 3-dimensional figures

cube: $6 s 2$ , where s is the length of the top side
rectangular box: $2 (\ell w + \ell h + w h)$ the length divided by height
sphere: $4π r 2$ , where π is the ratio of circumference to radius of a circle, 3.1415926535897932385..., 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 $(x,y)\in D\subset\mathbb{R}^2$ and $D$ is a region in the xy-plane with the smooth boundary:

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

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 $(u,v)\in D\subset\mathbb{R}^2$ :