Circumference at a Given Latitude

Date: 01/26/2001 at 17:52:23
From: Lynn
Subject: Circumference - at latitude of globe

I know the circumferance of the earth at the equator in miles. 
How can I calculate the circumference of a line around the earth (a 
circle) at 40 degrees north latitude?

Date: 01/26/2001 at 20:37:52
From: Doctor Fenton
Subject: Re: Circumference - at latitude of globe

Hi Lynn,

Thanks for writing to Dr. Math. 

If you're familiar with trigonometry, a point on the circle of 
latitude at latitude L has as its radius one leg of a triangle with 
center at the center of the Earth, one vertex at the point, and the 
other vertex on the polar axis: P is the point on the latitude circle, 
O is the center of the Earth, and Z is the point on the polar axis the 
same distance above the equatorial plane as P.

      polar axis
          |
          | radius at latitude L
        Z |---------P  
          |       /
          |      /
          |     /
          |    /
          |   /
          |  /
          | / L 
          |/_______________  Equatorial radius
          O

By geometry, Angle OPZ is also L, the same angle as the latitude.  OP
is the same as the equatorial radius, if you consider Earth a sphere, 
so 

     ZP
     -- = cos(L)
     OP

and

    ZP = OP*cos(L).

The circumference at latitude L is

    2*pi*ZP = 2*pi*OP*cos(L)

            = (equatorial circumference) * cos(L).

So, just multiply the equatorial circumference by the cosine of the
latitude, and you will have the circumference at that latitude.

If you have further questions, please write again.

- Doctor Fenton, The Math Forum
  http://mathforum.org/dr.math/