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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 for measuring area, with exact conversions, include:
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]}
(see Green's theorem)
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:
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>:
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.
Look up area in Wiktionary, the free dictionary. |
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