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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 term surface area refers to the total area of the exposed surface of a 3-dimensional solid, 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]}.
UnitsEdit
Units for measuring area include:
- area (a) = 100 square meters (m²)
- hectare (ha) = 100 ares (a) = 10000 square meters (m²)
- square kilometre (km²) = 100 hectars (ha) = 10000 ares (a) = 1000000 square metres (m²)
- square megametre (Mm²) = 10^{12} square metres
- square foot = 144 square inches = 0.09290304 square metres (m²)
- square yard = Template:Convert = 0.83612736 square metres (m²)
- square perch = 30.25 square yards = 25.2928526 square metres (m²)
- acre = 10 square chains (also one furlong by one chain); or 160 square perches; or 4840 square yards; or Template:Convert = 4046.8564224 square metres (m²)
- square mile = Template:Convert = 2.5899881103 square kilometers (km²)
FormulæEdit
Shape | Equation | Variables |
---|---|---|
Square | BH or LW | B = Base, H = Height or L = Length, W=Width |
Regular triangle (equilateral triangle) | BH/2( | $ s $ is the length of one side of the triangle. |
Regular hexagon | $ \frac{3\sqrt{3}}{2}s^2\,\! $ | $ s $ is the length of one side of the hexagon. |
Regular octagon | $ 2\left(1+\sqrt{2}\right)s^2\,\! $ | $ s $ is the length of one side of the octagon. |
Any regular polygon | $ \frac{1}{2}a p \,\! $ | $ a $ is the apothem, or the radius of an inscribed circle in the polygon, and $ p $ is the perimeter of the polygon. |
Any regular polygon | $ \frac{ns^2} {4 \cdot \tan(\pi/n)}\,\! $ | $ s $ is the sidelength and $ n $ is the number of sides. |
Any regular polygon (using degree measure) | $ \frac{ns^2} {4 \cdot \tan(180^\circ/n)}\,\! $ | $ s $ is the sidelength and $ n $ is the number of sides. |
Rectangle | $ lw \,\! $ | $ l $ and $ w $ are the lengths of the rectangle's sides (length and width). |
Parallelogram (in general) | $ bh\,\! $ | $ b $ and $ h $ are the length of the base and the length of the perpendicular height, respectively. |
Rhombus | $ \frac{1}{2}ab $ | $ a $ and $ b $ are the lengths of the two diagonals of the rhombus. |
Triangle | $ \frac{1}{2}bh \,\! $ | $ b $ and $ h $ are the base and altitude (measured perpendicular to the base), respectively. |
Triangle | $ \frac{1}{2} a b \sin(C)\,\! $ | $ a $ and $ b $ are any two sides, and $ C $ is the angle between them. |
Circle | $ \pi r^2\ \text{or}\ \frac{\pi d^2}{4} \,\! $ | $ r $ is the radius and $ d $ the diameter. |
Ellipse | $ \pi ab \,\! $ | $ a $ and $ b $ are the semi-major and semi-minor axes, respectively. |
Trapezoid | $ \frac{1}{2}(a+b)h \,\! $ | $ a $ and $ b $ are the parallel sides and $ h $ the distance (height) between the parallels. |
Total surface area of a Cylinder | $ 2\pi r^2+2\pi r h \,\! $ | $ r $ and $ h $ are the radius and height, respectively. |
Lateral surface area of a cylinder | $ 2 \pi r h \,\! $ | $ r $ and $ h $ are the radius and height, respectively. |
Total surface area of a Cone | $ \pi r (l + r) \,\! $ | $ r $ and $ l $ are the radius and slant height, respectively. |
Lateral surface area of a cone | $ \pi r l \,\! $ | $ r $ and $ l $ are the radius and slant height, respectively. |
Total surface area of a Sphere | $ 4\pi r^2\ \text{or}\ \pi d^2\,\! $ | $ r $ and $ d $ are the radius and diameter, respectively. |
Total surface area of an ellipsoid | See the article. | |
Circular sector | $ \frac{1}{2} r^2 \theta \,\! $ | $ r $ and $ \theta $ are the radius and angle (in radians), respectively. |
Square to circular area conversion | $ \frac{4}{\pi} A\,\! $ | $ A $ is the area of the square in square units. |
Circular to square area conversion | $ \frac{1}{4} C\pi\,\! $ | $ C $ 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 formulæEdit
Areas of 2-dimensional figuresEdit
- 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 (x_{1}y_{2}+ x_{2}y_{3}+ x_{3}y_{1} - x_{2}y_{1}- x_{3}y_{2}- x_{1}y_{3}) 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, (x_{1},y_{1}) (x_{2},y_{2}) (x_{3},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.
Area in calculusEdit
- 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. $
Surface area of 3-dimensional figures Edit
- cube: $ 6s^2 $, where s is the length of the top side
- rectangular box: $ 2 (\ell w + \ell h + w h) $ the length divided by height
- cone: $ \pi r\left(r + \sqrt{r^2 + h^2}\right) $, where r is the radius of the circular base, and h is the height. That can also be rewritten as $ \pi r^2 + \pi r l $ where r is the radius and l is the slant height of the cone. $ \pi r^2 $ is the base area while $ \pi r l $ is the lateral surface area of the cone.
- prism: 2 * Area of Base + Perimeter of Base * Height
General formulaEdit
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 $:
- $ A=\iint_D \left|\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}\right|\,du\,dv. $^{[1]}
Area minimisationEdit
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 alsoEdit
- Equi-areal mapping
- Integral
- Orders of magnitude (area)—A list of areas by size.
- Volume
ReferencesEdit
- ↑ ^{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 linksEdit
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