Area

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This article is about the geometric quantity. For other uses, see Area (disambiguation).

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The surface area of a 3-dimensional solid is the total area of the exposed surface, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential geometry of surfaces.[1]

Units

Units for measuring area, with exact conversions, include:

Formulae

Area.svg
Common formulae for area:
Shape Formula Variables
Regular triangle (equilateral triangle) <math>\tfrac14\sqrt{3}s^2\,\!</math> <math>s</math> is the length of one side of the triangle.
Triangle <math>\sqrt{s(s-a)(s-b)(s-c)}\,\!</math> <math> s </math> is half the perimeter, <math>a</math>, <math>b</math> and <math>c</math> are the length of each side.
Triangle <math>\tfrac12 a b \sin(C)\,\!</math> <math>a</math> and <math>b</math> are any two sides, and <math>C</math> is the angle between them.
Triangle <math>\tfrac12bh \,\!</math> <math>b</math> and <math>h</math> are the base and altitude (measured perpendicular to the base), respectively.
Square <math>s^2\,\!</math> <math>s</math> is the length of one side of the square.
Rectangle <math>lw \,\!</math> <math>l</math> and <math>w</math> are the lengths of the rectangle's sides (length and width).
Rhombus <math>\tfrac12ab</math> <math>a</math> and <math>b</math> are the lengths of the two diagonals of the rhombus.
Parallelogram <math>bh\,\!</math> <math>b</math> is the length of the base and <math>h</math> is the perpendicular height.
Trapezoid <math>\tfrac12(a+b)h \,\!</math> <math>a</math> and <math>b</math> are the parallel sides and <math>h</math> the distance (height) between the parallels.
Regular hexagon <math>\tfrac32\sqrt{3}s^2\,\!</math> <math>s</math> is the length of one side of the hexagon.
Regular octagon <math>2\left(1+\sqrt{2}\right)s^2\,\!</math> <math>s</math> is the length of one side of the octagon.
Regular polygon <math>\frac{ns^2} {4 \cdot \tan(\pi/n)}\,\!</math> <math> s </math> is the sidelength and <math>n</math> is the number of sides.
<math>\tfrac12a p \,\!</math> <math>a</math> is the apothem, or the radius of an inscribed circle in the polygon, and <math>p</math> is the perimeter of the polygon.
Circle <math>\pi r^2\ \text{or}\ \frac{\pi d^2}{4} \,\!</math> <math>r</math> is the radius and <math>d</math> the diameter.
Circular sector <math>\tfrac12 r^2 \theta \,\!</math> <math>r</math> and <math>\theta</math> are the radius and angle (in radians), respectively.
Ellipse <math>\pi ab \,\!</math> <math>a</math> and <math>b</math> are the semi-major and semi-minor axes, respectively.
Total surface area of a Cylinder <math>2\pi r (r + h)\,\!</math> <math>r</math> and <math>h</math> are the radius and height, respectively.
Lateral surface area of a cylinder <math>2 \pi r h \,\!</math> <math>r</math> and <math>h</math> are the radius and height, respectively.
Total surface area of a Cone <math>\pi r (r + l) \,\!</math> <math>r</math> and <math>l</math> are the radius and slant height, respectively.
Lateral surface area of a cone <math>\pi r l \,\!</math> <math>r</math> and <math>l</math> are the radius and slant height, respectively.
Total surface area of a Sphere <math>4\pi r^2\ \text{or}\ \pi d^2\,\!</math> <math>r</math> and <math>d</math> are the radius and diameter, respectively.
Total surface area of an ellipsoid   See the article.
Total surface area of a Pyramid <math>B+\frac{P L}{2}\,\!</math> <math>B</math> is the base area, <math>P</math> is the base perimeter and <math>L</math> is the slant height.
Square to circular area conversion <math>\frac{4}{\pi} A\,\!</math> <math>A</math> is the area of the square in square units.
Circular to square area conversion <math>\frac{1}{4} C\pi\,\!</math> <math>C</math> is the area of the circle in circular units.

The above calculations show how to find the area of many common shapes.

The area of irregular polygons can be calculated using the "Surveyor's formula".[2]

Additional formulae

Areas of 2-dimensional figures

  • a triangle: <math>\tfrac12Bh</math> (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: <math>\sqrt{s(s-a)(s-b)(s-c)}</math>(where a, b, c are the sides of the triangle, and <math>s = \tfrac12(a + b + c)</math> 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 (x1y2+ x2y3+ x3y1 - x2y1- x3y2- x1y3) 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). The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to use Infinitesimal calculus to find the area.
  • a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points: <math>i + \frac{b}{2} - 1</math>, where i is the number of grid points inside the polygon and b is the number of boundary points. This result is known as Pick's theorem.

Area in calculus

The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions
  • 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 <math> {1 \over 2} \int_0^{2\pi} r^2 \, d\theta </math>.
  • the area enclosed by a parametric curve <math>\vec u(t) = (x(t), y(t)) </math> with endpoints <math> \vec u(t_0) = \vec u(t_1) </math> is given by the line integrals
<math> \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 </math>

(see Green's theorem)

or the z-component of
<math>{1 \over 2} \oint_{t_0}^{t_1} \vec u \times \dot{\vec u} \, dt.</math>

Surface area of 3-dimensional figures

  • cube: <math>6s^2</math>, where s is the length of the top side
  • rectangular box: <math>2 (\ell w + \ell h + w h)</math> the length divided by height
  • cone: <math>\pi r\left(r + \sqrt{r^2 + h^2}\right)</math>, where r is the radius of the circular base, and h is the height. That can also be rewritten as <math>\pi r^2 + \pi r l </math> where r is the radius and l is the slant height of the cone. <math>\pi r^2 </math> is the base area while <math>\pi r l </math> is the lateral surface area of the cone.
  • prism: 2 × Area of Base + Perimeter of Base × Height

General formula

The general formula for the surface area of the graph of a continuously differentiable function <math>z=f(x,y),</math> where <math>(x,y)\in D\subset\mathbb{R}^2</math> and <math>D</math> is a region in the xy-plane with the smooth boundary:

<math> A=\iint_D\sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2+1}\,dx\,dy. </math>

Even more general formula for the area of the graph of a parametric surface in the vector form <math>\mathbf{r}=\mathbf{r}(u,v),</math> where <math>\mathbf{r}</math> is a continuously differentiable vector function of <math>(u,v)\in D\subset\mathbb{R}^2</math>:

<math> A=\iint_D \left|\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}\right|\,du\,dv. </math>[1]

Minimization

Given a wire contour, the surface of least area spanning ("filling") it is a minimal surface. Familiar examples include soap bubbles.

The question of the filling area of the Riemannian circle remains open.

See also

References

Notes

  1. 1.0 1.1 do Carmo, Manfredo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976. Page 98.
  2. http://www.maa.org/pubs/Calc_articles/ma063.pdf

External links

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