# Area Area[reg]

gives the area of the two-dimensional region reg.

Area[{x1,,xn},{s,smin,smax},{t,tmin,tmax}]

gives the area of the parametrized surface whose Cartesian coordinates xi are functions of s and t.

Area[{x1,,xn},{s,smin,smax},{t,tmin,tmax},chart]

interprets the xi as coordinates in the specified coordinate chart.

# Details and Options • Area is also known as surface area.
• A two-dimensional region can be embedded in any dimension greater than or equal to two.
• • In Area[x,{s,smin,smax},{t,tmin,tmax}], if x is a scalar, Area returns the area of the parametric surface {s,t,x}.
• Coordinate charts in the fourth argument of Area can be specified as triples {coordsys,metric,dim} in the same way as in the first argument of CoordinateChartData. The short form in which dim is omitted may be used.
• The following options can be given:
•  AccuracyGoal Infinity digits of absolute accurary sought Assumptions \$Assumptions assumptions to make about parameters GenerateConditions Automatic whether to generate conditions on parameters PerformanceGoal \$PerformanceGoal aspects of performance to try to optimize PrecisionGoal Automatic digits of precision sought WorkingPrecision Automatic the precision used in internal computations
• Symbolic limits of integration are assumed to be real and ordered. Symbolic coordinate chart parameters are assumed to be in range given by the "ParameterRangeAssumptions" property of CoordinateChartData.
• Area can be used with symbolic regions in GeometricScene.

# Examples

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## Basic Examples(4)

The area of a disk:

The surface area of a sphere:

The area of an annulus with inner radius 1 and outer radius 2:

The surface area of the open cylinder , expressed in cylindrical coordinates:

## Scope(21)

### Special Regions(5)

Area is defined for a 2D Simplex embedded in any dimension:

The area of a Polygon:

In 3D:

Disk:

Disk can be used as an ellipse:

The Area of a Sphere is its surface area:

### Formula Regions(2)

The area of a disk represented as an ImplicitRegion:

The surface area of a sphere:

The area of a disk represented as a ParametricRegion:

Using a rational parameterization of the disk:

The surface area of a sphere:

### Mesh Regions(2)

The area of a MeshRegion:

A 2D mesh embedded in 3D:

The area of a BoundaryMeshRegion:

### Derived Regions(3)

The area of a RegionIntersection:

The measure of a TransformedRegion:

The surface area of a RegionBoundary:

### Parametric Formulas(6)

The area of an ellipse with semimajor axes 2 and 1:

The area of a half-cone at in spherical coordinates:

The surface area of a torus of major radius 5 and minor radius 2:

The area of a "flat torus" embedded in four-dimensional space:

The area of the paraboloid over the rectangle :

The area of a surface in a three-sphere using stereographic coordinates:

### Geographic Regions(3)

The area of a polygon with GeoPosition:

The area of a polygon with GeoGridPosition:

Area works on polygons with geographic entities:

## Options(3)

### Assumptions(1)

The area of an ellipse with arbitrary semimajor axes and :

Adding an assumption that the semimajor axes are positive simplifies the answer:

### WorkingPrecision(2)

Compute the Area using machine arithmetic:

In some cases, the exact answer cannot be computed:

Find the Area using 30 digits of precision:

## Applications(4)

The area of a function surface :

The region is a surface:

Equivalently:

Find the average density of a region with density given by :

Compute the surface area of a polyhedron:

PolyhedronData gives a surface, not a solid, so Area can be used:

Compute the surface area of a Solomon seal knot:

The region is a surface in 3D:

## Properties & Relations(5)

Area is a non-negative quantity:

Area[r] is the same as for 2D regions:

Embedded in 3D:

Area[x,s,t,c] is equivalent to RegionMeasure[x,{s,t},c]:

For a 2D region, Area is the integral of 1 over the region:

Embedded in 3D:

To get the surface area of a 3D region, use RegionBoundary:

## Possible Issues(2)

The parametric form of Area computes the area of possibly multiple coverings:

The region version computes the area of the image:

The area of a region of dimension other than two is Undefined: