The measure for Point corresponds to counts:

Points can be used in any number of dimensions:

The measure for Line corresponds to arc length:

Lines can be used in any number of dimensions:

Rectangle can be used in 2D, and the measure corresponds to area:

Cuboid can be used in any number of dimensions:

A Simplex can correspond to a point, line, or triangle in 2D:

Simplices can be used in any number of dimensions:

The measure of a standard unit simplex in dimension :

Polygon represents an area:

In 3D:

Disk can be used in 2D:

Ball can be used in any dimension, and the measure is the generalized volume:

The measure of unit balls in dimension :

Disk as an ellipse can be used in 2D:

Ellipsoid can be used in any dimension:

Circle can be used in 2D:

Cylinder can be used in 3D:

Cone can be used in 3D:

The measure of a disk represented as an ImplicitRegion:

A cylinder volume:

The measure of a disk represented as a ParametricRegion:

Using a rational parametrization of disk:

A cylinder volume:

The measure of a MeshRegion in 2D:

In 3D:

The measure of a BoundaryMeshRegion:

In 3D:

The measure of a RegionIntersection:

The measure of a TransformedRegion:

The measure of a RegionBoundary:

The measure of a polygon of geographic entities:

Polygons with GeoPosition:

The measure of a polygon with GeoGridPosition:

Length of a circular arc:

An infinite curve in polar coordinates with finite length:

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

The volume of its interior:

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

The hypervolume of a 4-sphere embedded in five dimensions:

The hypervolume of the paraboloidal function graph over the unit hypercube:

The length of a curve "bouncing" between the poles on the unit sphere:

The area of the unit square in stereographic coordinates on the sphere:

The implicit region can represent both ellipses and hyperbolas:

Adding the assumption gives the length of an ellipse only:

The area of an ellipse with arbitrary semimajor axes and :

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

Compute the arc length using machine arithmetic:

Find the area using 30 digits of precision:

For point sets, the counting measure is used. Each point contributes 1 to the measure:

For constant point mass , multiply the measure by to get the total mass:

For a varying point mass function , use Integrate:

The length of a function curve :

The length of an implicit curve:

In 3D:

Find a formula for the length of a Peano curve:

Find the total charge along a wire with constant charge density :

For varying density , use Integrate:

The area of a function surface :

Total mass for a rectangular region:

With uniform mass density :

With varying mass density given by , use Integrate:

Total mass for a Ball with constant density :

For a varying density function , use Integrate:

Find the mass of ethanol in a Cone:

Density of ethanol:

Volume of cone:

Mass of ethanol in the cone:

Find the mass of a Cylinder with a nonuniform mass density defined by :

Density of cylinder:

Volume of cylinder:

Mass of cylinder:

Derive a formula for the region measure of an -dimensional unit ball:

The volume of the 3D hypersurface :