# What equation should I use to calculate the number of miles in a degree of longitude for a given latitude?

August 19, 2008 4:56 AM Subscribe

What equation can I use to calculate the number of miles in a degree of longitude for a given latitude?

Because the number of miles in a degree of longitude becomes zero at the poles, it's safe to say that as we travel north (anywhere in the USA, anyways) the number of miles in a degree of longitude shrinks...

Because the number of miles in a degree of longitude becomes zero at the poles, it's safe to say that as we travel north (anywhere in the USA, anyways) the number of miles in a degree of longitude shrinks...

On land (statute miles, as opposed to nautical) you're looking at more like 68.7 * cos(latitude)

posted by le morte de bea arthur at 5:10 AM on August 19, 2008

posted by le morte de bea arthur at 5:10 AM on August 19, 2008

If you're looking for an approximation, you can use Great Circle Distance, which assumes that the earth is a sphere.

If you want some quick approximations, its 69 miles at the equator, 0 at the pole, and about 49 miles at 45 degree latitude.

If you want something precise, try Google Maps or Google Earth with its distance calculation. This should take into account WGS84 or something similar

If you're looking for code, try the JScience or Sedris SDK.

posted by miasma at 5:12 AM on August 19, 2008

If you want some quick approximations, its 69 miles at the equator, 0 at the pole, and about 49 miles at 45 degree latitude.

If you want something precise, try Google Maps or Google Earth with its distance calculation. This should take into account WGS84 or something similar

If you're looking for code, try the JScience or Sedris SDK.

posted by miasma at 5:12 AM on August 19, 2008

Or if you want a quick approximations, google for "geodesic distance" ala http://www.movable-type.co.uk/scripts/latlong-vincenty.html

posted by miasma at 5:15 AM on August 19, 2008

posted by miasma at 5:15 AM on August 19, 2008

Best answer: This is basic trigonometry. I'll show you how.

If you imagine the Earth is a sphere, then what is the radius of a circle of constant latitude?

Draw a triangle to any point on that circle from the center of the Earth.

The hypotenus is just the radius of the Earth. We know the angle - its the latitude.

Remember SOHCAHTOA?

Well, Cosine(latitude)=Adjacent/Hypotenus

Adjacent is the radius of the latitude circle.

So, Adjacent=Cos(L)/R

You want to know the ratio of that circle to the circle at the equator, right?

Ratio=Cos(L)/Cos(0)=Cos(L)

So, at Latitude L, the length of a degree of longitude is Cos(L) times the length at the equator.

posted by vacapinta at 5:16 AM on August 19, 2008 [1 favorite]

If you imagine the Earth is a sphere, then what is the radius of a circle of constant latitude?

Draw a triangle to any point on that circle from the center of the Earth.

The hypotenus is just the radius of the Earth. We know the angle - its the latitude.

Remember SOHCAHTOA?

Well, Cosine(latitude)=Adjacent/Hypotenus

Adjacent is the radius of the latitude circle.

So, Adjacent=Cos(L)/R

You want to know the ratio of that circle to the circle at the equator, right?

Ratio=Cos(L)/Cos(0)=Cos(L)

So, at Latitude L, the length of a degree of longitude is Cos(L) times the length at the equator.

posted by vacapinta at 5:16 AM on August 19, 2008 [1 favorite]

Um, folks, cardboard gave the correct answer at first. No need to keep answering, unless to do as vacapinta did and explain the reasoning behind the answer.

posted by brianogilvie at 5:20 AM on August 19, 2008 [1 favorite]

posted by brianogilvie at 5:20 AM on August 19, 2008 [1 favorite]

The circumference of the Earth is roughly 25,000 miles, so 1 degree of longitude at the equator is approximately 25,000 miles/360 degrees = 69.4 miles. (It's 69.2 miles if you use the more exact value, 24,901.55 miles, for the circumference.)

69 miles*cos(latitude) should be close enough for government work.

posted by lukemeister at 5:43 AM on August 19, 2008

69 miles*cos(latitude) should be close enough for government work.

posted by lukemeister at 5:43 AM on August 19, 2008

Please do not look up "geodesic distance" or "great circle distance" or find the distance in google earth. They have nothing to do with the question you asked. The correct answer is quite simple, has been given several times, and involves the cosine of the latitude.

posted by exphysicist345 at 6:51 PM on August 19, 2008

posted by exphysicist345 at 6:51 PM on August 19, 2008

Well, because the earth is not perfectly spherical, the above answer is only approximate.

Here is a more accurate formula, which models the earth as an ellipsoid, rather than a sphere.

(Ultimately, there is no *exact* formula, because the earth is not an *exact* geometric shape)

posted by jpdoane at 8:36 AM on August 20, 2008

Here is a more accurate formula, which models the earth as an ellipsoid, rather than a sphere.

(Ultimately, there is no *exact* formula, because the earth is not an *exact* geometric shape)

posted by jpdoane at 8:36 AM on August 20, 2008

This thread is closed to new comments.

posted by cardboard at 5:06 AM on August 19, 2008