Area of a “rectangle” on a globe

What do we mean by rectangle?

This post will derive the area of a spherical region bounded by lines of latitude and longitude. Such a region corresponds to an actual rectangle on a Mercator projection map, with sides aligned with the coordinate axes, and is approximately a rectangle on a sphere if the rectangle is not too big.

What do we know up front?

Before we get into detailed equations, we know that the area will be proportional to the difference in longitude. If we're looking that the area between two parallels, such as the equator and the Arctic Circle, the area between 10 and 20 longitude is the same as the area between 80 and 90 longitude, and twice the area between 72 and 77 longitude.

The difficulty is latitude. Say we look at squares on a map that are 1 of longitude wide and 1 of latitude tall. Those squares are on the map will correspond to more area on the globe for latitudes near the equator, and less area at high latitudes.

So the area bounded by longitudes ₁ and ₂ and latitudes ₁ and ₂ will depend on ₁ and ₂ individually, but only on the difference ₁ - ₂.

Spherical caps

The region on a sphere above a fixed line of latitude is called a spherical cap. The northern hemisphere, the region above the equator, would be a very large spherical cap. The region inside the Arctic Circle would be a smaller spherical cap.

Let R be the radius of the earth. Then the surface area above a latitude is

A = 2R^2(1 - sin ).

You could derive this using calculus by thinking of the spherical cap as a surface of revolution.

Spherical bands

Given two latitudes ₁ and ₂ with latitudes ₁ > ₂, the area of a band between latitude ₁ and latitude ₂ is the area of the spherical cap above latitudes ₂ minus the area of the spherical cap above latitudes ₁. This gives

A = 2R^2(sin ₁ - sin ₂).

Area of latitude/longitude grid

Now we can find the area of the region bounded by longitudes ₁ and ₂ and latitudes ₁ and ₂. The total area between latitudes ₁ and ₂ is given by the equation above. The subset of this area between longitudes ₁ and ₂ is proportional to ₁ - ₂. If longitude is measured in radians then

A = R^2 (sin ₁ - sin ₂) (₁ - ₂).

If longitude is measured in degrees, we have

A = R^2 (sin ₁ - sin ₂) (₁ - ₂)/180.

Related posts

• Spherical distance
• Ellipsoidal distance
• Spherical trig in a nutshell

The post Area of a rectangle" on a globe first appeared on John D. Cook.