 # Area

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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.

## Units

Units for measuring surface area include:

Metric
square metre (m²) = SI derived unit
are (a) = 100 square metres (m²)
hectare (ha) = 10,000 square metres (m²)
square kilometre (km²) = 1,000,000 square metres (m²)
square megametre (Mm²) = 1012 square metres (m²)
US & Imperial Units
square foot = 144 square inches = 0.09290304 square metres (m²)
square yard = 9 square feet = 0.83612736 square metres (m²)
square perch = 30.25 square yards = 25.2928526 square metres (m²)
acre = 160 square perches or 4,840 square yards or 43,560 square feet = 4046.8564224 square metres (m²)
square mile = 640 acres = 2.5899881103 square kilometres (km²)

## Useful formulas

Common equations for area:
Shape Equation Variables
Square $s^2\,\!$ $s$ is the length of the side of the square.
Regular triangle $\frac{\sqrt{3}}{4}s^2\,\!$ $s$ 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 $2(1+\sqrt{2})s^2\,\!$ $s$ 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 $\frac{P^2/n} {4 \cdot tan(\pi/n)}\,\!$ $P$ is the Perimeter and $n$ is the number of sides.
Any regular polygon (using degree measure) $\frac{P^2/n} {4 \cdot tan(180^\circ/n)}\,\!$ $P$ is the Perimeter and $n$ is the number of sides.
Rectangle $l \cdot w \,\!$ $l$ and $w$ are the lengths of the rectangle's sides (length and width).
Parallelogram (in general) $b \cdot h\,\!$ $b$ 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 $\frac{1}{2}b \cdot h \,\!$ $b$ and $h$ are the base and altitude (measured perpendicular to the base), respectively.
Triangle $\frac{1}{2}\cdot a \cdot b \cdot sinC\,\!$ $a$ and $b$ are any two sides, and $C$ is the angle between them.
Circle $\pi r^2 \,\!$, or $\pi d^2/4 \,\!$ $r$ 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 $4\pi r^2\,\!$ 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.

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