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### 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
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/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Trigonometry

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