# How large is a Maidenhead field?

The Maidenhead locator system divides the earth into fields, squares, and subsquares. The first two characters in a Maidenhead locator specify the square. These are letters A through R representing 20 degrees of longitude or 10 degrees of latitude. Latitude A runs from the South Pole to 80° south of the equator. Latitude R runs from 80° north of the equator up to the North Pole.

The area of a field depends on latitude. The previous post was a lemma for this post, solving the general problem of finding the area of a region bounded by longitudes θ1 and θ2 and latitudes φ1 and φ2. Working in degrees, the area is

A = π r² (sin φ1 − sin φ2) (θ1 − θ2) / 180.

So area of the ith field above or below the equator is

A = π r² (sin 10− sin 10(i−1)° ) / 6.

Fields whose second character is I or J are nearest to the equator and have the largest area. This area is

π r² sin 10° / 6

If we take r to be 3959 miles (6371 km) this gives an area of about 950,000 square miles or 2,500,000 square kilometers.

Let’s do a quick check to see whether this result is reasonable. The circumference of the earth is about C = 40,000 km . The fields on either side of the equator are roughly Euclidean rectangles C/18 wide and C/36 tall. This also gives an area of about 2.5 million km².

For fields nearest the poles, fields with second character A or R, the area is

π r² (sin 90° − sin 80° ) / 6

which works out to be 83,000 square miles or 215,000 square kilometers.

Here’s a complete list of field areas based on the second letter.

```    |-------+----------+---------|
| Field | sq miles |   sq km |
|-------+----------+---------|
| A     |    83119 |  215250 |
| B     |   246831 |  639211 |
| C     |   403044 | 1043750 |
| D     |   547010 | 1416575 |
| E     |   674356 | 1746359 |
| F     |   781211 | 2023080 |
| G     |   864330 | 2238330 |
| H     |   921187 | 2385571 |
| I     |   950054 | 2460326 |
| J     |   950054 | 2460326 |
| K     |   921187 | 2385571 |
| L     |   864330 | 2238330 |
| M     |   781211 | 2023080 |
| N     |   674356 | 1746359 |
| O     |   547010 | 1416575 |
| P     |   403044 | 1043750 |
| Q     |   246831 |  639211 |
| R     |    83119 |  215250 |
|-------+----------+---------|
```

The area of a Maidenhead square is approximately 1/100 of the area of the field it is in. The area of a square also depends on latitude, but not as much as the area of a field does.

The area of a Maidenhead subsquare is approximately 1/24² = 1/576 of its square.

A typical Maidenhead field has area around 800,000 square miles or 2,000,000 square kilometers. (By “typical” I mean well populated areas. Much of the world’s population lives at around 30 or 40 degrees latitude.) So a typical square has area 8,000 square miles or 20,000 square kilometers. A typical subsquare will be around 14 square miles or 35 square kilometers.

## Related posts

 It isn’t a coincidence that this is such a round number. The meter was initially defined as 1/10,000,000 of the distance from the North Pole to the equator.