## Feet Per Minute

I recently have found a new hobby: Geocaching. I've been wondering just how precise a GPS receiver can get (assuming perfect reception). Latitude and Longitude are measured in degrees, latitude measured North or South from 0 to 90 degrees, longitude measured East or West from 0 to 180 degrees. Each degree is divided into 60 "minutes". Each minute is divided into 60 "seconds", though most displays of coordinates just break the minutes into three decimal places, like "N 38 degrees, 47.213 minutes". Assuming the Earth is a sphere with a circumference of 25000 miles, how many feet are in each minute? And thus, how precise is a thousandth of a minute?

The answer for latitude is constant, but the answer for longitude will vary with the latitude. (At the poles, no distance is traveled for a change in longitude.) Can you find a formula for both? Feel free to send URLs of web references, too.

Source: Original.

Solutions were received from Alan Zimmermann, Samantha Levin, and Joseph DeVincentis.
From Alan Zimmermann:

One thousandth of one minute of latitude is 25000 x 5280 / (360 x 60 x 1000) feet = 6 1/9 feet.

One thousandth of one minute of longitude is cos(latitude) * (6 1/9 feet).

For NYC, where I live and where the latitude is 40 degrees 43 minutes, .001 minutes of longitude is 4.63 feet.
In Chico, where the latitude is 39 degrees 44 minutes, .001 minutes of longitude is 4.70 feet.
At the Arctic Circle (66 degrees 34 minutes ), .001 minutes of longitude is 2.43 feet.
At the Tropic of Cancer (23 degrees, 27 minutes), .001 minutes of longitude is 5.61 feet.
At the Equator, .001 minutes of longitude is 6.11 feet.

From Samantha Levin:

First, consider the equator. If the circumference of the earth is 25000 miles, that is 132,000,000 feet (5,280 feet per mile). There are 360*60=21,600 minutes, so there are 132,000,000/21,600,000 = 6 1/9 feet in a thousandth of a minute. This will be the precision for latitude at the equator, and the precision for longitude everywhere.

Now consider the precision for latitude at latitude k. The radius of the circle going around the earth at latitude k is R*cos(k), where R is the radius of the earth at the equator. So the number of feet per minute at latitude k will be cos(k)*6 1/9.

From Joseph DeVincentis:

There are 360 degrees in a circle, times 60 minutes per degree, times 1000 thousandths per minute = 21,600,000 thousanths around the earth.

The circumference of the earth is 25000 miles times 5280 feet per mile, or 132 million feet.

Thus, 1/1000 minute of latitude, or 1/1000 minute of longitude at the equator, is equal to 6.11 feet, or 6 feet, 1 1/3 inches.

The length of 1/1000 minute of longitude shrinks at other latitudes in proportion to the cosine of the latitude. At 40 degrees latitude (a mid-US latitude), it's 4.68 feet. At 60 degrees latitude (southern Alaska) it's 3.06 feet. And as noted, at the pole it vanishes to zero.

Of course, there's a point after which the accuracy of the measurement is less than the accuracy of the display. Since this only depends on latitude which the device calculates anyway, it's possible for the device to start dropping digits from the longitude when it can no longer give any meaningful information with them. Without knowing the true (absolute distance) precision, it's not possible to know when this occurs, but if we assume that precision is 6.11 feet, then the last digit would lose all meaning at 84.3 degrees latitude. (That is, at that position a hundredth of a minute of longitude is 6.11 feet.) Given that, I doubt they bothered with any such programming.

Mail to Ken