Area
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]}
Contents
Units
Units for measuring area include:
- are (a) = 100 square metres (m²)
- hectare (ha) = 100 ares (a) = 10000 square metres
- square kilometre (km²) = 100 hectares (ha) = 10000 ares = 1000000 square metres
- square megametre (Mm²) = 10^{12} square metres
- square foot = 144 square inches = 0.09290304 square metres
- square yard = 9 square feet (0.84 m^{2}) = 0.83612736 square metres
- square perch = 30.25 square yards = 25.2928526 square metres
- acre = 10 square chains (also one furlong by one chain); or 160 square perches; or 4840 square yards; or 43,560 square feet (4,047 m^{2}) = 4046.8564224 square metres
- square mile = 640 acres (2.6 km^{2}) = 2.5899881103 square kilometers
Formulæ
Shape | Equation | Variables |
---|---|---|
Regular triangle (equilateral triangle) | is the length of one side of the triangle. | |
Triangle | and are any two sides, and is the angle between them. | |
Triangle | and are the base and altitude (measured perpendicular to the base), respectively. | |
Square | is the length of one side of the square. | |
Rectangle | and are the lengths of the rectangle's sides (length and width). | |
Rhombus | and are the lengths of the two diagonals of the rhombus. | |
Parallelogram | and are the length of the base and the length of the perpendicular height, respectively. | |
Trapezoid | and are the parallel sides and the distance (height) between the parallels. | |
Regular hexagon | is the length of one side of the hexagon. | |
Regular octagon | is the length of one side of the octagon. | |
Regular polygon | is the sidelength and is the number of sides. | |
is the apothem, or the radius of an inscribed circle in the polygon, and is the perimeter of the polygon. | ||
Circle | is the radius and the diameter. | |
Circular sector | and are the radius and angle (in radians), respectively. | |
Ellipse | and are the semi-major and semi-minor axes, respectively. | |
Total surface area of a Cylinder | and are the radius and height, respectively. | |
Lateral surface area of a cylinder | and are the radius and height, respectively. | |
Total surface area of a Cone | and are the radius and slant height, respectively. | |
Lateral surface area of a cone | and are the radius and slant height, respectively. | |
Total surface area of a Sphere | and are the radius and diameter, respectively. | |
Total surface area of an ellipsoid | See the article. | |
Square to circular area conversion | is the area of the square in square units. | |
Circular to square area conversion | 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 formulæ
Areas of 2-dimensional figures
- 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: (where a, b, c are the sides of the triangle, and 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.
- 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: , 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 .
- the area enclosed by a parametric curve with endpoints is given by the line integrals
(see Green's theorem)
- or the z-component of
Surface area of 3-dimensional figures
- cube: , where s is the length of the top side
- rectangular box: the length divided by height
- cone: , where r is the radius of the circular base, and h is the height. That can also be rewritten as where r is the radius and l is the slant height of the cone. is the base area while 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 where and is a region in the xy-plane with the smooth boundary:
Even more general formula for the area of the graph of a parametric surface in the vector form where is a continuously differentiable vector function of :
- ^{[1]}
Area minimisation
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.
- Volume
References
- ↑ ^{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. |