Given a latitude how many miles is the corresponding longitude?

Question

OK so lines of longitude (the distance/circumference around the earth horizontally) differ based on what latitude you are at (0 at north and south poles up to ~25k at the equator.)

So given a latitude, how can I determine how many miles it would be to go directly east/west around the earth until I was back at the starting point.

If it is easier mathematically to treat the earth like a perfect sphere, then that is fine. I do not need it to be precise, just close enough.

Answer 1

The parallel of latitude is actually a circle of radius $r\cos(\alpha)$

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Thus, the length of the parallel of latitude $\alpha$ is $2\pi r\cos(\alpha)$, where $r$ is the radius of the Earth.

Answer 2

Consider this image:

The north pole is to the right. Let $\alpha$ represent your latitude, ranging from $0$ to $\frac{\pi}{2}$ radians. Then $\sin\alpha$ represents the radius of the circle you're interested in. You want to scale this up to earth-size, so multiply everything by $r$, the radius of the earth. Then, the circle you seek has circumference $2\pi r \sin\alpha$.