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.

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
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 ${\displaystyle s^{2}\,\!}$ ${\displaystyle s}$ is the length of the side of the square.
Regular triangle ${\displaystyle {\frac {\sqrt {3}}{4}}s^{2}\,\!}$ ${\displaystyle s}$ is the length of one side of the triangle.
Regular hexagon ${\displaystyle {\frac {3{\sqrt {3}}}{2}}s^{2}\,\!}$ ${\displaystyle s}$ is the length of one side of the hexagon.
Regular octagon ${\displaystyle 2(1+{\sqrt {2}})s^{2}\,\!}$ ${\displaystyle s}$ is the length of one side of the octagon.
Any regular polygon ${\displaystyle {\frac {1}{2}}ap\,\!}$ ${\displaystyle a}$ is the apothem, or the radius of an inscribed circle in the polygon, and ${\displaystyle p}$ is the perimeter of the polygon.
Any regular polygon ${\displaystyle {\frac {P^{2}/n}{4\cdot \tan(\pi /n)}}\,\!}$ ${\displaystyle P}$ is the Perimeter and ${\displaystyle n}$ is the number of sides.
Any regular polygon (using degree measure) ${\displaystyle {\frac {P^{2}/n}{4\cdot \tan(180^{\circ }/n)}}\,\!}$ ${\displaystyle P}$ is the Perimeter and ${\displaystyle n}$ is the number of sides.
Rectangle ${\displaystyle lw\,\!}$ ${\displaystyle l}$ and ${\displaystyle w}$ are the lengths of the rectangle's sides (length and width).
Parallelogram (in general) ${\displaystyle bh\,\!}$ ${\displaystyle b}$ and ${\displaystyle h}$ are the length of the base and the length of the perpendicular height, respectively.
Rhombus ${\displaystyle {\frac {1}{2}}ab}$ ${\displaystyle a}$ and ${\displaystyle b}$ are the lengths of the two diagonals of the rhombus.
Triangle ${\displaystyle {\frac {1}{2}}bh\,\!}$ ${\displaystyle b}$ and ${\displaystyle h}$ are the base and altitude (measured perpendicular to the base), respectively.
Triangle ${\displaystyle {\frac {1}{2}}ab\sin C\,\!}$ ${\displaystyle a}$ and ${\displaystyle b}$ are any two sides, and ${\displaystyle C}$ is the angle between them.
Circle ${\displaystyle \pi r^{2},\,\!}$ or ${\displaystyle \pi d^{2}/4\,\!}$ ${\displaystyle r}$ is the radius and ${\displaystyle d}$ the diameter.
Ellipse ${\displaystyle \pi ab\,\!}$ ${\displaystyle a}$ and ${\displaystyle b}$ are the semi-major and semi-minor axes, respectively.
Trapezoid ${\displaystyle {\frac {1}{2}}(a+b)h\,\!}$ ${\displaystyle a}$ and ${\displaystyle b}$ are the parallel sides and ${\displaystyle h}$ the distance (height) between the parallels.
Total surface area of a Cylinder ${\displaystyle 2\pi r^{2}+2\pi rh\,\!}$ ${\displaystyle r}$ and ${\displaystyle h}$ are the radius and height, respectively.
Lateral surface area of a cylinder ${\displaystyle 2\pi rh\,\!}$ ${\displaystyle r}$ and ${\displaystyle h}$ are the radius and height, respectively.
Total surface area of a Cone ${\displaystyle \pi r(l+r)\,\!}$ ${\displaystyle r}$ and ${\displaystyle l}$ are the radius and slant height, respectively.
Lateral surface area of a cone ${\displaystyle \pi rl\,\!}$ ${\displaystyle r}$ and ${\displaystyle l}$ are the radius and slant height, respectively.
Total surface area of a Sphere ${\displaystyle 4\pi r^{2}\,\!}$ or ${\displaystyle \pi d^{2}\,\!}$ ${\displaystyle r}$ and ${\displaystyle d}$ are the radius and diameter, respectively.
Total surface area of an ellipsoid   See the article.
Circular sector ${\displaystyle {\frac {1}{2}}r^{2}\theta \,\!}$ ${\displaystyle r}$ and ${\displaystyle \theta }$ are the radius and angle (in radians), respectively.
Square to circular area conversion ${\displaystyle {\frac {4}{\pi }}A\,\!}$ ${\displaystyle A}$ is the area of the square in square units.
Circular to square area conversion ${\displaystyle {\frac {1}{4}}C\pi \,\!}$ ${\displaystyle C}$ is the area of the circle in circular units.

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