## 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 [1].

## 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 θ_{1} and θ_{2} and latitudes φ_{1} and φ_{2} will depend on φ_{1} and φ_{2} individually, but only on the difference θ_{1} − θ_{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 φ is

*A* = 2π*R*²(1 − sin φ).

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

## Spherical bands

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

*A* = 2π*R*²(sin φ_{1} − sin φ_{2}).

## Area of latitude/longitude grid

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

*A* = *R*² (sin φ_{1} − sin φ_{2}) (θ_{1} − θ_{2}).

If longitude is measured in **degrees**, we have

*A* = π *R*² (sin φ_{1} − sin φ_{2}) (θ_{1} − θ_{2})/180.

## Related posts

[1] As Nathan helpfully pointed out in the comments, it might be clearer to say “not too big and not too close to one of the poles.” Another way to put it might be to say the idea of what is “big” depends on how close to a pole you are.

So my first thought was that it should be approximately (θ1 − θ2) * (φ1 – φ2) * cos((φ1 + φ2) / 2), as long as the two latitudes aren’t to far apart, because the length of a line of latitude is proportional to cos(φ).

The exact answer comes up as involving sin(φ1) – sin(φ2). But hey, what happens if we take a limit? Let δ = φ1 – φ2; then sin(φ2 + δ) – sin(φ2) / δ is a difference quotient, and when δ becomes small, it turns into the derivative of sin, which is of course cos.

(More to the point, but less fun, the spherical cap formula has sin in it because sin is the antiderivative of cos.)

Surely the proportionality constant in the degree case is π/180, the size of 1 degree in radians. Or am I missing something subtle?

Or is it that the proportionality constant for the radian case is 1/(2π)? The units you want are fractions of a whole circle, because we’re interested in some fraction of the whole cap rather than the radian or degree angle between the longitudes.

Thanks, Tom. I updated the post to make that more clear.

I think it would be more accurate to say “…and is approximately a rectangle on a sphere if the rectangle is not too big and not too close to the poles.”

For example, a circle of radius 1cm around the North Pole is a “rectangle” by this definition.