Area Explained

For other uses see Area (disambiguation).

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.

Units

Units for measuring area include:

are (a) = 100 square meters (m²)

hectare (ha) = 100 ares (a) = 10000 square meters (m²)

square kilometre (km²) = 100 hectars (ha) = 10000 ares (a) = 1000000 square metres (m²)

square megametre (Mm²) = 10¹² square metres

square foot = 144 square inches = 0.09290304 square metres (m²)

square yard = 9square feet = 0.83612736 square metres (m²)

square perch = 30.25 square yards = 25.2928526 square metres (m²)

acre = 10 square chains (also one furlong by one chain); or 160 square perches; or 4840 square yards; or 43560square feet = 4046.8564224 square metres (m²)

square mile = 640acres = 2.5899881103 square kilometers (km²)

Formulae

Common formulae for area: ! Shape! Equation! Variables

Square

s²

is the length of one side of the square.

Regular triangle (equilateral triangle)

	\sqrt{3

}

is the length of one side of the triangle.

Regular hexagon

	3\sqrt{3

}

is the length of one side of the hexagon.

Regular octagon

2\left(1+\sqrt{2}\right)s²

is the length of one side of the octagon.

Any regular polygon

	1
	2

is the apothem, or the radius of an inscribed circle in the polygon, and

is the perimeter of the polygon.

Any regular polygon

	ns²
	4 ⋅ \tan(\pi/n)

is the sidelength and

is the number of sides.

Any regular polygon (using degree measure)

	ns²
	4 ⋅ \tan(180^\circ/n)

is the sidelength and

is the number of sides.

Rectangle

and

are the lengths of the rectangle's sides (length and width).

Parallelogram (in general)

and

are the length of the base and the length of the perpendicular height, respectively.

Rhombus

	1
	2

and

are the lengths of the two diagonals of the rhombus.

Triangle

	1
	2

and

are the base and altitude (measured perpendicular to the base), respectively.

Triangle

	1
	2

ab\sin(C)

and

are any two sides, and

is the angle between them.

Circle

\pi

2 or	\pid²
	4

is the radius and

the diameter.

Ellipse

\piab

and

are the semi-major and semi-minor axes, respectively.

Trapezoid

	1
	2

(a+b)h

and

are the parallel sides and

the distance (height) between the parallels.

Total surface area of a Cylinder

2\pir^2+2\pirh

and

are the radius and height, respectively.

Lateral surface area of a cylinder

2\pirh

and

are the radius and height, respectively.

Total surface area of a Cone

\pir(l+r)

and

are the radius and slant height, respectively.

Lateral surface area of a cone

\pirl

and

are the radius and slant height, respectively.

Total surface area of a Sphere

4\pir^2 or \pid²

and

are the radius and diameter, respectively.

Total surface area of an ellipsoid

See the article.

Circular sector

	1
	2

r²\theta

and

\theta

are the radius and angle (in radians), respectively.

Square to circular area conversion

	4
	\pi

is the area of the square in square units.

Circular to square area conversion

	1
	4

C\pi

is the area of the circle in circular units.

A revolution of f(x) about the x-axis

2\pi

	b
\int
	a

f(x)

\sqrt

Area of surface of revolution of f(x) about the y-axis

2\pi

	b
\int
	a

\sqrt

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

The area of irregular polygons can be calculated using the "Surveyor's formula".^[1]

How to define area

Area is a quantity expressing the size of the contents of a region on a 2-dimensional surface. Points and lines have zero area, cf. space-filling curves. A region may have infinite area, for example the entire Euclidean plane. The 3-dimensional analog of area is volume. Although area seems to be one of the basic notions in geometry, it is not easy to define even in the Euclidean plane. Most textbooks avoid defining an area, relying on self-evidence. For polygons in the Euclidean plane, one can proceed as follows:

The area of a polygon in the Euclidean plane is a positive number such that:

The area of the unit square is equal to one.
Congruent polygons have equal areas.
(additivity) If a polygon is a union of two polygons which do not have common interior points, then its area is the sum of the areas of these polygons.

It remains to show that the notion of area thus defined does not depend on the way one subdivides a polygon into smaller parts.

A typical way to introduce area is through the more advanced notion of Lebesgue measure. In the presence of the axiom of choice it is possible to prove the existence of shapes whose Lebesgue measure cannot be meaningfully defined. Such 'shapes' (they cannot a fortiori be simply visualised) enter into Tarski's circle-squaring problem (and, moving to three dimensions, in the Banach–Tarski paradox). The sets involved do not arise in practical matters.

In three dimensions, the analog of area is called volume. The n dimensional analog, usually referred to as 'content', is defined by means of a measure or as a Lebesgue integral.

Additional formulas

Areas of 2-dimensional figures

a triangle:

	Bh
	2

(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:

\sqrt{s(s-a)(s-b)(s-c)}

(where a, b, c are the sides of the triangle, and

	a+b+c
	2

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 (x₁y₂+ x₂y₃+ x₃y₁ - x₂y₁- x₃y₂- x₁y₃) 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, (x₁,y₁) (x₂,y₂) (x₃,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.

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

{1\over2}

	2\pi
\int
	0

r²d\theta

the area enclosed by a parametric curve

\vecu(t)=(x(t),y(t))

with endpoints

\vecu(t₀₎=\vecu(t₁₎

is given by the line integrals

	t₁
\oint
	t₀

•

dt=-

	t₁
\oint
	t₀

•

dt={1\over2}

	t₁
\oint
	t₀

•

-y

•

(see Green's theorem)

or the z-component of

{1\over2}

	t₁
\oint
	t₀

\vecu x

•

\vecu

dt.

Surface area of 3-dimensional figures

cube

6s²

, where s is the length of the top side

rectangular box

2(\ellw+\ellh+wh)

the length divided by height

cone

\pir\left(r+\sqrt{r²+h^2}\right)

, where r is the radius of the circular base, and h is the height. That can also be rewritten as

\pir²+\pirl

where r is the radius and l is the slant height of the cone.

\pir²

is the base area while

\pirl

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

z=f(x,y),

where

(x,y)\inD\subsetR²

and

is a region in the xy-plane with the smooth boundary:

A=\iint

D\sqrt{\left(	\partialf
	\partialx

2+\left(	\partialf
	\partialy

\right)

\right)^2+1}dxdy.

Even more general formula for the area of the graph of a parametric surface in the vector form

r=r(u,v),

where

is a continuously differentiable vector function of

(u,v)\inD\subsetR²

A=\iint_D\left|

	\partialr	x
	\partialu

	\partialr
	\partialv

\right|dudv.

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.

External links

Notes and References

http://www.maa.org/pubs/Calc_articles/ma063.pdf