Area

Published on Monday, February 6th 2006. Edited by Foo Lowgagsit, Kiel, Germany.

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 θ 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.

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