Area is the quantity that expresses the extent
of a two-dimensional figure or shape, or planar
lamina, in the plane. Surface area is its
analog on the two-dimensional surface of a
three-dimensional object. Area can be understood
as the amount of material with a given thickness
that would be necessary to fashion a model
of the shape, or the amount of paint necessary
to cover the surface with a single coat. It
is the two-dimensional analog of the length
of a curve (a one-dimensional concept) or
the volume of a solid (a three-dimensional
concept).
The area of a shape can be measured by comparing
the shape to squares of a fixed size. In the
International System of Units (SI), the standard
unit of area is the square metre (written
as m2), which is the area of a square whose
sides are one metre long. A shape with an
area of three square metres would have the
same area as three such squares. In mathematics,
the unit square is defined to have area one,
and the area of any other shape or surface
is a dimensionless real number.
There are several well-known formulas for
the areas of simple shapes such as triangles,
rectangles, and circles. Using these formulas,
the area of any polygon can be found by dividing
the polygon into triangles. For shapes with
curved boundary, calculus is usually required
to compute the area. Indeed, the problem of
determining the area of plane figures was
a major motivation for the historical development
of calculus.For a solid shape such as a sphere,
cone, or cylinder, the area of its boundary
surface is called the surface area. Formulas
for the surface areas of simple shapes were
computed by the ancient Greeks, but computing
the surface area of a more complicated shape
usually requires multivariable calculus.
Area plays an important role in modern mathematics.
In addition to its obvious importance in geometry
and calculus, area is related to the definition
of determinants in linear algebra, and is
a basic property of surfaces in differential
geometry. In analysis, the area of a subset
of the plane is defined using Lebesgue measure,
though not every subset is measurable. In
general, area in higher mathematics is seen
as a special case of volume for two-dimensional
regions.Area can be defined through the use
of axioms, defining it as a function of a
collection of certain plane figures to the
set of real numbers. It can be proved that
such a function exists.
== Formal definition ==
An approach to defining what is meant by "area"
is through axioms. "Area" can be defined as
a function from a collection M of special
kind of plane figures (termed measurable sets)
to the set of real numbers which satisfies
the following properties:
For all S in M, a(S) ≥ 0.
If S and T are in M then so are S ∪ T and
S ∩ T, and also a(S∪T) = a(S) + a(T) − a(S∩T).
If S and 
T are in M with S ⊆ T then T − S is in
M and a(T−S) = a(T) − a(S).
If a set S is in M and S is congruent to T
then T is also in M and a(S) = a(T).
Every rectangle R is 
in M. If the rectangle has length h and breadth
k then a(R) = hk.
Let Q be a set enclosed between two step regions
S and T. A step region is formed from a finite
union of adjacent rectangles resting on a
common base, i.e. S ⊆ Q ⊆ T. If there
is a unique number c such that a(S) ≤ c
≤ a(T) for all such step regions S and T,
then a(Q) = c.It can be proved that such an
area function actually exists.
== Units ==
Every unit of length has a corresponding unit
of area, namely the area of a square with
the given side length. Thus areas can be measured
in square metres (m2), square centimetres
(cm2), square millimetres (mm2), square kilometres
(km2), square feet (ft2), square yards (yd2),
square miles (mi2), and so forth. Algebraically,
these units can be thought of as the squares
of the corresponding length units.
The SI unit of area is the square metre, which
is considered an SI derived unit.
=== Conversions ===
Calculation of the area of a square whose
length and width are 1 metre would be:
1 metre x 1 metre = 1 m2and so, a rectangle
with different sides (say length of 3 metres
and width of 2 metres) would have an area
in square units that can be calculated as:
3 metres x 2 metres = 6 m2. This is equivalent
to 6 million square millimetres. Other useful
conversions are:
1 square kilometre = 1,000,000 square metres
1 square metre = 10,000 square centimetres
= 1,000,000 square millimetres
1 square centimetre = 100 square millimetres.
==== Non-metric units ====
In non-metric units, the conversion between
two square units is the square of the conversion
between the corresponding length units.
1 foot = 12 inches,the relationship between
square feet and square inches is
1 square foot = 144 square inches,where 144
= 122 = 12 × 12. Similarly:
1 square yard = 9 square feet
1 square mile = 3,097,600 square yards = 27,878,400
square feetIn addition, conversion factors
include:
1 square inch = 6.4516 square centimetres
1 square foot = 0.09290304 square metres
1 square yard = 0.83612736 square metres
1 square mile = 2.589988110336 square kilometres
=== Other units including historical ===
There are several other common units for area.
The are was the original unit of area in the
metric system, with:
1 are = 100 square metresThough the are has
fallen out of use, the hectare is still commonly
used to measure land:
1 hectare = 100 ares = 10,000 square metres
= 0.01 square kilometresOther uncommon metric
units of area include the tetrad, the hectad,
and the myriad.
The acre is also commonly used to measure
land areas, where
1 acre = 4,840 square yards = 43,560 square
feet.An acre is approximately 40% of a hectare.
On the atomic scale, area is measured in units
of barns, such that:
1 barn = 10−28 square meters.The barn is
commonly used in describing the cross-sectional
area of interaction in nuclear physics.In
India,
20 dhurki = 1 dhur
20 dhur = 1 khatha
20 khata = 1 bigha
32 khata = 1 acre
== History ==
=== Circle area ===
In the 5th century BCE, Hippocrates of Chios
was the first to show that the area of a disk
(the region enclosed by a circle) is proportional
to the square of its diameter, as part of
his quadrature of the lune of Hippocrates,
but did not identify the constant of proportionality.
Eudoxus of Cnidus, also in the 5th century
BCE, also found that the area of a disk is
proportional to its radius squared.Subsequently,
Book I of Euclid's Elements dealt with equality
of areas between two-dimensional figures.
The mathematician Archimedes used the tools
of Euclidean geometry to show that the area
inside a circle is equal to that of a right
triangle whose base has the length of the
circle's circumference and whose height equals
the circle's radius, in his book Measurement
of a Circle. (The circumference is 2πr, and
the area of a triangle is half the base times
the height, yielding the area πr2 for the
disk.) Archimedes approximated the value of
π (and hence the area of a unit-radius circle)
with his doubling method, in which he inscribed
a regular triangle in a circle and noted its
area, then doubled the number of sides to
give a regular hexagon, then repeatedly doubled
the number of sides as the polygon's area
got closer and closer to that of the circle
(and did the same with circumscribed polygons).
Swiss scientist Johann Heinrich Lambert in
1761 proved that π, the ratio of a circle's
area to its squared radius, is irrational,
meaning it is not equal to the quotient of
any two whole numbers. In 1794 French mathematician
Adrien-Marie Legendre proved that π2 is irrational;
this also proves that π is irrational. In
1882, German mathematician Ferdinand von Lindemann
proved that π is transcendental (not the
solution of any polynomial equation with rational
coefficients), confirming a conjecture made
by both Legendre and Euler.
=== Triangle area ===
Heron (or Hero) of Alexandria found what is
known as Heron's formula for the area of a
triangle in terms of its sides, and a proof
can be found in his book, Metrica, written
around 60 CE. It has been suggested that Archimedes
knew the formula over two centuries earlier,
and since Metrica is a collection of the mathematical
knowledge available in the ancient world,
it is possible that the formula predates the
reference given in that work.In 499 Aryabhata,
a great mathematician-astronomer from the
classical age of Indian mathematics and Indian
astronomy, expressed the area of a triangle
as one-half the base times the height in the
Aryabhatiya (section 2.6).
A formula equivalent to Heron's was discovered
by the Chinese independently of the Greeks.
It was published in 1247 in Shushu Jiuzhang
("Mathematical Treatise in Nine Sections"),
written by Qin Jiushao.
=== Quadrilateral area ===
In the 7th century CE, Brahmagupta developed
a formula, now known as Brahmagupta's formula,
for the area of a cyclic quadrilateral (a
quadrilateral inscribed in a circle) in terms
of its sides. In 1842 the German mathematicians
Carl Anton Bretschneider and Karl Georg Christian
von Staudt independently found a formula,
known as Bretschneider's formula, for the
area of any quadrilateral.
=== General polygon area ===
The development of Cartesian coordinates by
René Descartes in the 17th century allowed
the development of the surveyor's formula
for the area of any polygon with known vertex
locations by Gauss in the 19th century.
=== Areas determined using calculus ===
The development of integral calculus in the
late 17th century provided tools that could
subsequently be used for computing more complicated
areas, such as the area of an ellipse and
the surface areas of various curved three-dimensional
objects.
== Area formulas ==
=== Polygon formulas ===
For a non-self-intersecting (simple) polygon,
the Cartesian coordinates
(
x
i
,
y
i
)
{\displaystyle (x_{i},y_{i})}
(i=0, 1, ..., n-1) of whose n vertices are
known, the area is given by the surveyor's
formula:
A
=
1
2
|
∑
i
=
0
n
−
1
(
x
i
y
i
+
1
−
x
i
+
1
y
i
)
|
{\displaystyle A={\frac {1}{2}}|\sum _{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i})|}
where when i=n-1, then i+1 is expressed as
modulus n and so refers to 0.
==== Rectangles ====
The most basic area formula is the formula
for the area of a rectangle. Given a rectangle
with length l and width w, the formula for
the area is:
A = lw (rectangle).That is, the area of the
rectangle is the length multiplied by the
width. As a special case, as l = w in the
case of a square, the area of a square with
side length s is given by the formula:
A = s2 (square).The formula for the area of
a rectangle follows directly from the basic
properties of area, and is sometimes taken
as a definition or axiom. On the other hand,
if geometry is developed before arithmetic,
this formula can be used to define multiplication
of real numbers.
==== Dissection, parallelograms, and triangles
====
Most other simple formulas for area follow
from the method of dissection.
This involves cutting a shape into pieces,
whose areas must sum to the area of the original
shape.
For an example, any parallelogram can be subdivided
into a trapezoid and a right triangle, as
shown in figure to the left. If the triangle
is moved to the other side of the trapezoid,
then the resulting figure is a rectangle.
It follows that the area of the parallelogram
is the same as the area of the rectangle:
A = bh (parallelogram).However, the same parallelogram
can also 
be cut along a diagonal into two congruent
triangles, as shown in the figure to the right.
It follows that the area of each triangle
is half the area of the parallelogram:
A
=
1
2
b
h
{\displaystyle A={\frac {1}{2}}bh}
(triangle).Similar arguments can be used to
find area formulas for the trapezoid as well
as more complicated polygons.
=== Area of curved shapes ===
==== Circles ====
The formula for the area of a circle (more
properly called the area enclosed by a circle
or the 
area of a disk) is based on a similar method.
Given a circle of radius r, it is possible
to partition the circle into sectors, as shown
in the figure to the right. Each sector is
approximately triangular in shape, and the
sectors can be rearranged to form an approximate
parallelogram. The height of this parallelogram
is r, and the width is half the circumference
of the circle, or πr. Thus, the total area
of the circle is πr2:
A = πr2 (circle).Though the dissection used
in this formula is only approximate, the error
becomes smaller and smaller as the circle
is partitioned into more and more sectors.
The limit of the areas of the approximate
parallelograms is exactly πr2, which is the
area of the circle.This argument is actually
a simple application of the ideas of calculus.
In ancient times, the method of exhaustion
was used in a similar way to find the area
of the circle, and this method is now recognized
as a precursor to integral calculus. Using
modern methods, the area of a circle can be
computed using a definite integral:
A
=
2
∫
−
r
r
r
2
−
x
2
d
x
=
π
r
2
.
{\displaystyle A\;=\;2\int _{-r}^{r}{\sqrt
{r^{2}-x^{2}}}\,dx\;=\;\pi r^{2}.}
==== Ellipses ====
The formula for the area enclosed by an ellipse
is related to the formula of a circle; for
an ellipse with semi-major and semi-minor
axes x and y the formula is:
A
=
π
x
y
.
{\displaystyle A=\pi xy.}
==== Surface area ====
Most basic formulas for surface area can be
obtained by cutting surfaces and flattening
them out. For example, if the side surface
of a cylinder (or any prism) is cut lengthwise,
the surface can be flattened out into a rectangle.
Similarly, if a cut is made along the side
of a cone, the side surface can be flattened
out into a sector of a circle, and the resulting
area computed.
The formula for the surface area of a sphere
is more difficult to derive: because a sphere
has nonzero Gaussian curvature, it cannot
be flattened out. The formula for the surface
area of a sphere was first obtained by Archimedes
in his work On the Sphere and Cylinder. The
formula is:
A = 4πr2 (sphere),where r is the radius of
the sphere. As with the formula for the area
of a circle, any derivation of this formula
inherently uses methods similar to calculus.
=== General formulas ===
==== Areas of 2-dimensional figures ====
A triangle:
1
2
B
h
{\displaystyle {\tfrac {1}{2}}Bh}
(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:
s
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
{\displaystyle {\sqrt {s(s-a)(s-b)(s-c)}}}
where a, b, c are the sides of the triangle,
and
s
=
1
2
(
a
+
b
+
c
)
{\displaystyle s={\tfrac {1}{2}}(a+b+c)}
is half of its perimeter. If an angle and
its two included sides are given, the area
is
1
2
a
b
sin
⁡
(
C
)
{\displaystyle {\tfrac {1}{2}}ab\sin(C)}
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
1
2
(
x
1
y
2
+
x
2
y
3
+
x
3
y
1
−
x
2
y
1
−
x
3
y
2
−
x
1
y
3
)
{\displaystyle {\tfrac {1}{2}}(x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{1}y_{3})}
. 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), and (x3,y3).
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 calculus to find the area.
A simple polygon constructed on a grid of
equal-distanced points (i.e., points with
integer coordinates) such that all the polygon's
vertices are grid points:
i
+
b
2
−
1
{\displaystyle i+{\frac {b}{2}}-1}
, where i is the number of grid points inside
the polygon and b is the number of boundary
points. This result is known as Pick's theorem.
==== Area in calculus ====
The area between a positive-valued curve and
the horizontal axis, measured between two
values a and b (b is defined as the larger
of the two values) on the horizontal axis,
is given by the integral from a to b of the
function that represents the curve:
A
=
∫
a
b
f
(
x
)
d
x
.
{\displaystyle A=\int _{a}^{b}f(x)\,dx.}
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):
A
=
∫
a
b
(
f
(
x
)
−
g
(
x
)
)
d
x
,
{\displaystyle A=\int _{a}^{b}(f(x)-g(x))\,dx,}
where
f
(
x
)
{\displaystyle f(x)}
is the curve with the greater y-value.An area
bounded by a function r = r(θ) expressed
in polar coordinates is:
A
=
1
2
∫
r
2
d
θ
.
{\displaystyle A={1 \over 2}\int r^{2}\,d\theta
.}
The area enclosed by a parametric curve
u
→
(
t
)
=
(
x
(
t
)
,
y
(
t
)
)
{\displaystyle {\vec {u}}(t)=(x(t),y(t))}
with endpoints
u
→
(
t
0
)
=
u
→
(
t
1
)
{\displaystyle {\vec {u}}(t_{0})={\vec {u}}(t_{1})}
is given by the line integrals:
∮
t
0
t
1
⁡
x
y
˙
d
t
=
−
∮
t
0
t
1
⁡
y
x
˙
d
t
=
1
2
∮
t
0
t
1
⁡
(
x
y
˙
−
y
x
˙
)
d
t
{\displaystyle \oint _{t_{0}}^{t_{1}}x{\dot
{y}}\,dt=-\oint _{t_{0}}^{t_{1}}y{\dot {x}}\,dt={1
\over 2}\oint _{t_{0}}^{t_{1}}(x{\dot {y}}-y{\dot
{x}})\,dt}
(see Green's theorem) or the z-component of
1
2
∮
t
0
t
1
⁡
u
→
×
u
→
˙
d
t
.
{\displaystyle {1 \over 2}\oint _{t_{0}}^{t_{1}}{\vec
{u}}\times {\dot {\vec {u}}}\,dt.}
==== Bounded area between two quadratic functions
====
To find the bounded area between two quadratic
functions, we subtract one from the other
to write the difference as
f
(
x
)
−
g
(
x
)
=
a
x
2
+
b
x
+
c
=
a
(
x
−
α
)
(
x
−
β
)
{\displaystyle f(x)-g(x)=ax^{2}+bx+c=a(x-\alpha
)(x-\beta )}
where f(x) is the quadratic upper bound and
g(x) is the quadratic lower bound. Define
the discriminant of f(x)-g(x) as
Δ
=
b
2
−
4
a
c
.
{\displaystyle \Delta =b^{2}-4ac.}
By simplifying the integral formula between
the graphs of two functions (as given in the
section above) and using Vieta's formula,
we can obtain
A
=
Δ
Δ
6
a
2
=
a
6
(
β
−
α
)
3
,
a
≠
0.
{\displaystyle A={\frac {\Delta {\sqrt {\Delta
}}}{6a^{2}}}={\frac {a}{6}}(\beta -\alpha
)^{3},\qquad a\neq 0.}
The above remains valid if one of the bounding
functions is linear instead of quadratic.
==== Surface area of 3-dimensional figures
====
Cone:
π
r
(
r
+
r
2
+
h
2
)
{\displaystyle \pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)}
, where r is the radius of the circular base,
and h is the height. That can also be rewritten
as
π
r
2
+
π
r
l
{\displaystyle \pi r^{2}+\pi rl}
or
π
r
(
r
+
l
)
{\displaystyle \pi r(r+l)\,\!}
where r is the radius and l is the slant height
of the cone.
π
r
2
{\displaystyle \pi r^{2}}
is the base area while
π
r
l
{\displaystyle \pi rl}
is the lateral surface area of the cone.
cube:
6
s
2
{\displaystyle 6s^{2}}
, where s is the length of an edge.
cylinder:
2
π
r
(
r
+
h
)
{\displaystyle 2\pi r(r+h)}
, where r is the radius of a base and h is
the height. The 2
π
{\displaystyle \pi }
r can also be rewritten as
π
{\displaystyle \pi }
d, where d is the diameter.
prism: 2B + Ph, where B is the area of a base,
P is the perimeter of a base, and h is the
height of the prism.
pyramid:
B
+
P
L
2
{\displaystyle B+{\frac {PL}{2}}}
, where B is the area of the base, P is the
perimeter of the base, and L is the length
of the slant.
rectangular prism:
2
(
ℓ
w
+
ℓ
h
+
w
h
)
{\displaystyle 2(\ell w+\ell h+wh)}
, where
ℓ
{\displaystyle \ell }
is the length, w is the width, and h is the
height.
==== General formula for surface area ====
The general formula for the surface area of
the graph of a continuously differentiable
function
z
=
f
(
x
,
y
)
,
{\displaystyle z=f(x,y),}
where
(
x
,
y
)
∈
D
⊂
R
2
{\displaystyle (x,y)\in D\subset \mathbb {R}
^{2}}
and
D
{\displaystyle D}
is a region in the xy-plane with the smooth
boundary:
A
=
∬
D
(
∂
f
∂
x
)
2
+
(
∂
f
∂
y
)
2
+
1
d
x
d
y
.
{\displaystyle A=\iint _{D}{\sqrt {\left({\frac
{\partial f}{\partial x}}\right)^{2}+\left({\frac
{\partial f}{\partial y}}\right)^{2}+1}}\,dx\,dy.}
An even more general formula for the area
of the graph of a parametric surface in the
vector form
r
=
r
(
u
,
v
)
,
{\displaystyle \mathbf {r} =\mathbf {r} (u,v),}
where
r
{\displaystyle \mathbf {r} }
is a continuously differentiable vector function
of
(
u
,
v
)
∈
D
⊂
R
2
{\displaystyle (u,v)\in D\subset \mathbb {R}
^{2}}
is:
A
=
∬
D
|
∂
r
∂
u
×
∂
r
∂
v
|
d
u
d
v
.
{\displaystyle A=\iint _{D}\left|{\frac {\partial
\mathbf {r} }{\partial u}}\times {\frac {\partial
\mathbf {r} }{\partial v}}\right|\,du\,dv.}
=== List of formulas ===
The above calculations show how to find the
areas of many common shapes.
The areas of irregular polygons can be calculated
using the "Surveyor's formula".
=== Relation of area to perimeter ===
The isoperimetric inequality states that,
for a closed curve of length L (so the region
it encloses has perimeter L) and for area
A of the region that it encloses,
4
π
A
≤
L
2
,
{\displaystyle 4\pi A\leq L^{2},}
and equality holds if and only if the curve
is a circle. Thus a circle has the largest
area of any closed figure with a given perimeter.
At the other extreme, a figure with given
perimeter L could have an arbitrarily small
area, as illustrated by a rhombus that is
"tipped over" arbitrarily far so that two
of its angles are arbitrarily close to 0°
and the other two are arbitrarily close to
180°.
For a circle, the ratio of the area to the
circumference (the term for the perimeter
of a circle) equals half the radius r. This
can be seen from the area formula πr2 and
the circumference formula 2πr.
The area of a regular polygon is half its
perimeter times the apothem (where the apothem
is the distance from the center to the nearest
point on any side).
=== Fractals ===
Doubling the edge lengths of a polygon multiplies
its area by four, which is two (the ratio
of the new to the old side length) raised
to the power of two (the dimension of the
space the polygon resides in). But if the
one-dimensional lengths of a fractal drawn
in two dimensions are all doubled, the spatial
content of the fractal scales by a power of
two that is not necessarily an integer. This
power is called the fractal dimension of the
fractal.
== Area bisectors ==
There are an infinitude of lines that bisect
the area of a triangle. Three of them are
the medians of the triangle (which connect
the sides' midpoints with the opposite vertices),
and these are concurrent at the triangle's
centroid; indeed, they are the only area bisectors
that go through the centroid. Any line through
a triangle that splits both the triangle's
area and its perimeter in half goes through
the triangle's incenter (the center of its
incircle). There are either one, two, or three
of these for any given triangle.
Any line through the midpoint of a parallelogram
bisects the area.
All area bisectors of a circle or other ellipse
go through the center, and any chords through
the center bisect the area. In the case of
a circle they are the diameters of the circle.
== Optimization ==
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.The circle has the largest
area of any two-dimensional object having
the same perimeter.
A cyclic polygon (one inscribed in a circle)
has the largest area of any polygon with a
given number of sides of the same lengths.
A version of the isoperimetric inequality
for triangles states that the triangle of
greatest area among all those with a given
perimeter is equilateral.The triangle of largest
area of all those inscribed in a given circle
is equilateral; and the triangle of smallest
area of all those circumscribed around a given
circle is equilateral.The ratio of the area
of the incircle to the area of an equilateral
triangle,
π
3
3
{\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}
, is larger than that of any non-equilateral
triangle.The ratio of the area to the square
of the perimeter of an equilateral triangle,
1
12
3
,
{\displaystyle {\frac {1}{12{\sqrt {3}}}},}
is larger than that for any other triangle.
== See also ==
Brahmagupta quadrilateral, a cyclic quadrilateral
with integer sides, integer diagonals, and
integer area.
Equi-areal mapping
Heronian triangle, a triangle with integer
sides and integer area.
List of triangle inequalities#Area
One-seventh area triangle, an inner triangle
with one-seventh the area of the reference
triangle.Routh's theorem, a generalization
of the one-seventh area triangle.Orders of
magnitude (area)—A list of areas by size.
Pentagon#Derivation of the area formula
Planimeter, an instrument for measuring small
areas, e.g. on maps.
Quadrilateral#Area of a convex quadrilateral
Robbins pentagon, a cyclic pentagon whose
side lengths and area are all rational numbers
