Why is the distance between Boulder and Philadelphia is 7 miles? Am I doing my math wrong?

This is the lat/long for Philadelphia: http://www.rcn.montana.edu/resources/tools/coordinates.aspx?nav=11&c=DD&md=24&mdt=Internationa开发者_开发问答l(1924)-Hayford(1909)&lat=39.947648&lath=N&lon=-75.151978&lonh=W

This is the lat/long for Boulder: http://www.rcn.montana.edu/resources/tools/coordinates.aspx?nav=11&c=DD&md=24&mdt=International(1924)-Hayford(1909)&lat=40.0149856&lath=N&lon=-105.2705456&lonh=W

That lat and long are correct (You can check it in Google Maps). UTM_east and UTM_north are also correct for both.

Now, plug the UTMs into the distance formula here: http://www.basic-mathematics.com/distance-formula-calculator.html

And you will get distance in meters, which is 7 miles.

Why on earth is Boulder 7 miles away from Philadelphia?

You can't just plug in the UTM coordinates like that because these two cities are not in the same UTM Zone.

EDIT:

And, as everyone else has pointed out, even if they were in the same zone, you shouldn't just apply a planar, cartesian distance calculation to the UTM coordinates because the UTM coordinates are based on a cylindrical projection. I was just pointing out that the largest contributing factor to your error was the zone issue.

Latitude and Longitude are a spherical coordinate system and the formula you're using only works on a plane. You need to use the haversine formula.

When I calculate the distance between the two points as if they were on a standard Cartesian plane, I get a distance of 29.9, which is really close to the tool's result:

The distance between these two points is 29.900202340452488

First, using a Cartesian distance calculator on a spherical object isn't going to give good results. :) (Leaving aside that the Earth isn't spherical, but it sure isn't flat either.)

BUT, let's assume for a second that using Cartesian distance is "good enough", the results here are measured in whatever units we input. And knowing that 1 degree is roughly 111 km, we get a quick guess that the distance between Boulder and Philadelphia is roughly 3318.9 km. Given that Google's driving directions between the two is roughly 2841 km, you can immediately see why applying Cartesian distance algorithms won't work on a sphere, and why you need to use the haversine formula.