For large regions (e.g. “rectangular” regions with sides greater than several hundred kilometres) surface areas will be noticeably affected by the curvature of the Earth. Such computations apply at large state and continental levels, or where the areas of ocean, ice or cloud cover are required over large regions. Computation of areas using spherical trigonometry or numerical integration is preferable in such cases.

The spherical area of a rectangular region defined by fixed latitude and longitude coordinates (e.g. determined by 100°W30°N and 90°W40°N) is greater than the area bounded by lines of constant latitude — the latter are not great circles, although the lines of longitude are. This effect increases rapidly nearer the poles and is weakest close to the equator. The MATLab Mapping Toolbox (MMT) function AREAQUAD provides a convenient way of computing the area of a latitude-longitude “quadrangle”, although it can be computed without too much difficulty using spherical trigonometry. The area north of a line of latitude, φ, is the key factor in the calculation (the surface area of an entire sphere is 4πR²). The former area is:

where R is the radius of the spherical Earth, e.g. 6378.137 kms. The area of the quadrangle is then simply the difference between the areas, A₁ and A₂, north of each of the two lines of latitude, φ₁ and φ₂, adjusted by the proportion of the Earth included in the difference in longitude values, λ₁ and λ₂:

A (simple) polygon with n vertices on the surface of the Earth has sides that are great circles and area:

where again R is the radius of the Earth and the θ_i are the internal angles of the polygon in radians measured at each polygon vertex. The simplest spherical polygon is a triangle (n=3 so n‑2=1) and on a sphere the sum of its internal angles must be greater than 180° (π), so the formula equates to 0 when the triangle is vanishingly small. A spherical triangle with internal angles that are all 180° is the largest possible, and is itself a great circle. In this case the formula yields A=2πR², as expected.

All polygon arcs must be represented by great circles and internal angles are to be measured with respect to these curved arcs. The term in brackets is sometimes known as the spherical excess and denoted with the letter E, thus A=ER². Note that every simple polygon on a sphere divides it into two sections, one smaller than the other unless all vertices lie on a great circle. If only the latitude and longitude values are known, the internal angles must be computed by calculating the true great circle bearing (or azimuth) from each vertex to the next — this can be achieved using spherical trigonometry or, for example, the MMT function AZIMUTH, which supports explicit geoid values. Azimuth values are reported from due north, so computations of internal angles need to adjust for this factor. Alternatively the MMT function AREAINT may be used, ideally with high point density (i.e. introducing additional vertices along the polygon arcs). This latter function computes the enclosed area numerically. The open source GIS, GRASS, provides a number of similar functions, including the source code modules area_poly1, which computes polygon areas on an ellipsoid where connecting lines are grid lines rather than great circles, and area_poly2 which computes planimetric polygon area.