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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:
- square metre (m2)
- are (a) = 100 square metres (m2)
- hectare (ha) = 100 ares = 10000 square metres
- square kilometre (km2) = 100 hectares = 10000 ares = 1000000 square metres
- square megametre (Mm2) = 1000000000000 square metres
- square foot = 144 square inches = 0.09290304 square metres
- square yard = 9 square feet = 0.83612736 square metres
- square perch = 30.25 square yards = 25.2928526 square metres
- acre = 10 square chains = one furlong by one chain = 160 square perches = 4840 square yards = 43560 square feet = 4046.8564224 square metres
- square mile = 640 acres = 2.589988110336 square kilometres
- Kanal (unit) = 0.125 acres.
Formulae
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 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
- Equi-areal mapping
- Integral
- Orders of magnitude (area)—A list of areas by size.
- Perimeter
- Volume
References
Notes
- ↑ 1.0 1.1 do Carmo, Manfredo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976. Page 98.
- ↑ http://www.maa.org/pubs/Calc_articles/ma063.pdf
External links
Look up area in Wiktionary, the free dictionary. |
- Weisstein, Eric W., "Area" from MathWorld.
- Area formulas
- Conversion cable diameter to circle cross-sectional area and vice versa
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