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Area is a quantity
Quantity

Quantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with Quality , substance, change, and relation....
 expressing the two-dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al size of a defined part of a surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
, typically a region bounded by a closed curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
. 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
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
. Area is an important invariant
Invariant

Invariant and invariance may have several meanings, among which are:* Invariant , an expression whose value doesn't change during execution ...
 in the differential geometry of surfaces
Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with smooth manifold surfaces with various additional structures, most often, a Riemannian metric....
.

s for measuring area include:

are
Are

Are is a unit of area, equal to 100 square metres , used for measuring land area. It was defined by older forms of the metric system, but is now outside of the modern SI ....
 (a) = 100 square meters (mē) hectare
Hectare

A hectare is a unit of area equal to , or one square hectometre , and commonly used for surveying.The hectare is used in most countries around the world, especially in domains concerned with land ownership, land planning, and land management, including law , agriculture, forestry, and town planning....
 (ha) = 100 ares (a) = 10000 square meters (mē) square kilometre
Square kilometre

Square kilometre , symbol km2, is a decimal multiple of the SI Units of measurement of surface area, the square metre, one of the SI derived units....
 (kmē) = 100 hectars (ha) = 10000 ares (a) = 1000000 square metres (mē) square megametre (Mmē) = 1012 square metres square foot
Square foot

The square foot is an imperial unit / U.S. customary unit of area, used mainly in the United States and United Kingdom. It is defined as the area of a Square with sides of 1 foot in length....
 = 144 square inches = 0.09290304 square metres (mē) square yard
Square yard

The square yard is an Imperial unit/U.S. customary unit unit of area, formerly used in most of the English language-speaking world but now generally replaced by the square metre outside of the US....
 = = 0.83612736 square metres (mē) square perch = 30.25 square yards = 25.2928526 square metres (mē) acre
Acre

The acre is a Units of measurement of area in a number of different systems, including the Imperial unit#Measures of area and United States customary units#Units of area systems....
 = 10 square chains (also one furlong
Furlong

A furlong is a measure of distance in imperial units and U.S. customary units. It is equal to one-eighth of a mile, 220 yards, 660 foot or 201.168 meters....
 by one chain); or 160 square perches; or 4840 square yards; or = 4046.8564224 square metres (mē) square mile
Square mile

The square mile is an Imperial system and US customary system of measure for an area equal to the area of a square of one mile. It should not be confused with miles square, which refers to the number of miles on each side squared....
 = = 2.5899881103 square kilometers (kmē)

le class="prettytable">
 Common formulae for area:
Shape Equation Variables
Square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
  is the length of one side of the square.
Regular triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 (equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
) 		is the length of one side of the triangle.
Regular hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
  is the length of one side of the hexagon.
Regular octagon
Octagon

In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....
  is the length of one side of the octagon.
Any regular polygon  is the apothem
Apothem

The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides....
, or the radius of an inscribed circle in the polygon, and is the perimeter of the polygon.
Any regular polygon  is the sidelength and is the number of sides.
Any regular polygon (using degree measure)  is the sidelength and is the number of sides.
Rectangle
Rectangle

In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
  and are the lengths of the rectangle's sides (length and width).
Parallelogram
Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
 (in general)		  and are the length of the base and the length of the perpendicular height, respectively.
Rhombus
Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
  and are the lengths of the two diagonals of the rhombus.
Triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
  and are the base
Base (geometry)

The base of any geometric figure is any side that you wish to measure from or, , any face that you wish to measure from. Bases are most commonly used in geometric formulas for area and volume....
 and altitude
Altitude (triangle)

In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side or an extension of the opposite side....
 (measured perpendicular to the base), respectively.
Triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
  and are any two sides, and is the angle between them.
Circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
  is the radius and the diameter
Diameter

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle....
.
Ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
  and are the semi-major
Semi-major axis

In geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae....
 and semi-minor
Semi-minor axis

In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis....
 axes, respectively.
Trapezoid
Trapezoid

In geometry, a trapezoid or trapezium is a quadrilateral with twoparallel sides. The term “trapezoid” is used in North America, while the term “trapezium” is prevalent in Britain....
  and are the parallel sides and the distance (height) between the parallels.
Total surface area of a Cylinder
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
  and are the radius and height, respectively.
Lateral surface area of a cylinder  and are the radius and height, respectively.
Total surface area of a Cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
  and are the radius and slant height
Slant height

The slant height of a right circular cone is the distance from any point on the circle to the apex of the cone.The slant height of a cone is given by the formula , where is the radius of the circle and is the height from the center of the circle to the apex of the cone....
, respectively.
Lateral surface area of a cone  and are the radius and slant height, respectively.
Total surface area of a Sphere  and are the radius and diameter, respectively.
Total surface area of an ellipsoid See the article.
Circular sector
Circular sector

A circular sector or circle sector, is the portion of a circle enclosed by two radius and an Arc , where the smaller area is known as the minor sector and the larger being the major sector....
  and are the radius and angle (in radians), respectively.
Square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 to circular area conversion		  is the area of the square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 in square units.
Circular
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 to square area conversion		  is the area of the circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 in circular units.


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

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

rea is a quantity expressing the size of the contents of a region on a 2-dimensional surface.






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Encyclopedia


Area is a quantity
Quantity

Quantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with Quality , substance, change, and relation....
 expressing the two-dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al size of a defined part of a surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
, typically a region bounded by a closed curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
. 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
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
. Area is an important invariant
Invariant

Invariant and invariance may have several meanings, among which are:* Invariant , an expression whose value doesn't change during execution ...
 in the differential geometry of surfaces
Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with smooth manifold surfaces with various additional structures, most often, a Riemannian metric....
.

Units

Units for measuring area include:

are
Are

Are is a unit of area, equal to 100 square metres , used for measuring land area. It was defined by older forms of the metric system, but is now outside of the modern SI ....
 (a) = 100 square meters (mē) hectare
Hectare

A hectare is a unit of area equal to , or one square hectometre , and commonly used for surveying.The hectare is used in most countries around the world, especially in domains concerned with land ownership, land planning, and land management, including law , agriculture, forestry, and town planning....
 (ha) = 100 ares (a) = 10000 square meters (mē) square kilometre
Square kilometre

Square kilometre , symbol km2, is a decimal multiple of the SI Units of measurement of surface area, the square metre, one of the SI derived units....
 (kmē) = 100 hectars (ha) = 10000 ares (a) = 1000000 square metres (mē) square megametre (Mmē) = 1012 square metres square foot
Square foot

The square foot is an imperial unit / U.S. customary unit of area, used mainly in the United States and United Kingdom. It is defined as the area of a Square with sides of 1 foot in length....
 = 144 square inches = 0.09290304 square metres (mē) square yard
Square yard

The square yard is an Imperial unit/U.S. customary unit unit of area, formerly used in most of the English language-speaking world but now generally replaced by the square metre outside of the US....
 = = 0.83612736 square metres (mē) square perch = 30.25 square yards = 25.2928526 square metres (mē) acre
Acre

The acre is a Units of measurement of area in a number of different systems, including the Imperial unit#Measures of area and United States customary units#Units of area systems....
 = 10 square chains (also one furlong
Furlong

A furlong is a measure of distance in imperial units and U.S. customary units. It is equal to one-eighth of a mile, 220 yards, 660 foot or 201.168 meters....
 by one chain); or 160 square perches; or 4840 square yards; or = 4046.8564224 square metres (mē) square mile
Square mile

The square mile is an Imperial system and US customary system of measure for an area equal to the area of a square of one mile. It should not be confused with miles square, which refers to the number of miles on each side squared....
 = = 2.5899881103 square kilometers (kmē)

Formulae

 Common formulae for area:
Shape Equation Variables
Square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
  is the length of one side of the square.
Regular triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 (equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
) 		is the length of one side of the triangle.
Regular hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
  is the length of one side of the hexagon.
Regular octagon
Octagon

In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....
  is the length of one side of the octagon.
Any regular polygon  is the apothem
Apothem

The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides....
, or the radius of an inscribed circle in the polygon, and is the perimeter of the polygon.
Any regular polygon  is the sidelength and is the number of sides.
Any regular polygon (using degree measure)  is the sidelength and is the number of sides.
Rectangle
Rectangle

In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
  and are the lengths of the rectangle's sides (length and width).
Parallelogram
Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
 (in general)		  and are the length of the base and the length of the perpendicular height, respectively.
Rhombus
Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
  and are the lengths of the two diagonals of the rhombus.
Triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
  and are the base
Base (geometry)

The base of any geometric figure is any side that you wish to measure from or, , any face that you wish to measure from. Bases are most commonly used in geometric formulas for area and volume....
 and altitude
Altitude (triangle)

In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side or an extension of the opposite side....
 (measured perpendicular to the base), respectively.
Triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
  and are any two sides, and is the angle between them.
Circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
  is the radius and the diameter
Diameter

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle....
.
Ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
  and are the semi-major
Semi-major axis

In geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae....
 and semi-minor
Semi-minor axis

In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis....
 axes, respectively.
Trapezoid
Trapezoid

In geometry, a trapezoid or trapezium is a quadrilateral with twoparallel sides. The term “trapezoid” is used in North America, while the term “trapezium” is prevalent in Britain....
  and are the parallel sides and the distance (height) between the parallels.
Total surface area of a Cylinder
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
  and are the radius and height, respectively.
Lateral surface area of a cylinder  and are the radius and height, respectively.
Total surface area of a Cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
  and are the radius and slant height
Slant height

The slant height of a right circular cone is the distance from any point on the circle to the apex of the cone.The slant height of a cone is given by the formula , where is the radius of the circle and is the height from the center of the circle to the apex of the cone....
, respectively.
Lateral surface area of a cone  and are the radius and slant height, respectively.
Total surface area of a Sphere  and are the radius and diameter, respectively.
Total surface area of an ellipsoid See the article.
Circular sector
Circular sector

A circular sector or circle sector, is the portion of a circle enclosed by two radius and an Arc , where the smaller area is known as the minor sector and the larger being the major sector....
  and are the radius and angle (in radians), respectively.
Square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 to circular area conversion		  is the area of the square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 in square units.
Circular
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 to square area conversion		  is the area of the circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 in circular units.


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

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

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
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
. 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:
  1. The area of the unit square
    Unit square

    The unit square is a square with all of the side lengths equalling 1....
     is equal to one.
  2. Congruent
    Congruence (geometry)

    In geometry, two sets of point are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translation s, rotations and reflection s....
     polygons have equal areas.
  3. (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
Lebesgue measure

In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space....
. In the presence of the axiom of choice
Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinite set many bins and there is no "rule" for which object t...
 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
Tarski's circle-squaring problem

Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a Disk in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area....
 (and, moving to three dimensions, in the Banach–Tarski paradox
Banach–Tarski paradox

The Banach?Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into several non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball....
). The sets involved do not arise in practical matters.

In three dimensions, the analog of area is called volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
. The n dimensional analog, usually referred to as 'content', is defined by means of a measure
Measure (mathematics)

In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset....
 or as a Lebesgue integral.

Additional formulas


Areas of 2-dimensional figures

  • a triangle: (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
    Heron's formula

    In geometry, Heron's formula states that the area of a triangle whose sides have lengths a, b, and c iswhere s is the semiperimeter of the triangle:...
     can be used: (where a, b, c are the sides of the triangle, and 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
    Shoelace formula

    The shoelace formula, or shoelace algorithm, is a mathematical algorithm to determine the area of a polygon whose vertices are described by ordered pairs in the plane....
     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
    Infinitesimal calculus

    Infinitesimal calculus was independently invented by both Gottfried Leibniz and Isaac Newton in the 1660s, drawing on the work of such mathematicians as Isaac Barrow and Rene Descartes....
     to find the area.


Area in calculus

Areabetweentwographs
*the area between the graph
Graph of a function

In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian coordinate system, together with Cartesian axes, etc....
s of two functions is equal
Equality (mathematics)

Equality is the paradigmatic example of the more general concept of equivalence relations on a set: those binary relations which are reflexive relation, symmetric relation, and transitive relation....
 to the integral
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
 of one function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
, f(x), minus
Subtraction

Subtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with....
 the integral of the other function, g(x).
  • an area bounded by a function r = r(?) expressed in polar coordinates is .
  • the area enclosed by a parametric curve with endpoints is given by the line integral
    Line integral

    In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use....
    s
(see Green's theorem
Green's theorem

In physics and mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C....
)
or the z-component of




Surface area of 3-dimensional figures

  • cube: , where s is the length of the top side
  • rectangular box: the length divided by height
  • cone
    Cone (geometry)

    A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
    : , where r is the radius of the circular base, and h is the height. That can also be rewritten as where r is the radius and l is the slant height of the cone. is the base area while is the lateral surface area of the cone.
  • prism
    Prism (geometry)

    In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
    : 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 where and is a region in the xy-plane with the smooth boundary:
Even more general formula for the area of the graph of a parametric surface
Parametric surface

A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters. Parametric representation is the most general way to specify a surface....
 in the vector form where is a continuously differentiable vector function of :


Area minimisation

Given a wire contour, the surface of least area spanning ("filling") it is a minimal surface
Minimal surface

In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
. Familiar examples include soap bubble
Soap bubble

A soap bubble is a very thin film of soap water that forms a sphere with an iridescence surface. Soap bubbles usually last for only a few moments before bursting: either on their own or on contact with another object....
s.

The question of the filling area
Filling area conjecture

In mathematics, in Riemannian geometry, Mikhail Gromov's filling area conjecture asserts that among all possible fillings of the Riemannian circle of length 2p by a surface with the strongly isometric property, the round hemisphere has the least area....
 of the Riemannian circle
Riemannian circle

In metric space theory and Riemannian geometry, the term Riemannian circle refers to a great circle equipped with its great-circle distance. In more detail, the term refers to the circle equipped with its intrinsic Riemannian metric of a compact 1-dimensional manifold of total length 2π, as opposed to the extrinsic metric obtaine...
 remains open.

See also

  • Equi-areal mapping
    Real matrices (2 x 2)

    The 2 x 2 real matrices are the linear mappings of the Cartesian coordinate system into itself by the ruleThe set of all such real matrices is denoted by M....
  • Integral
    Integral

    Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
  • Orders of magnitude (area)
    Orders of magnitude (area)

    This page is a progressive and labeled list of the SI area orders of magnitude, with certain examples appended to some list objects....
    —A list of areas by size.
  • Volume
    Volume

    The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....


External links