# Latitude "length" formula?

For simplification, let’s just pretend the Earth is 24000 miles around at the equator. So the “length” of 0º is 24000 miles.

What is the formula for the “length” of the other latitudes?

I think I’ve found the answer for 90º North, and I should soon have the answer for 90º South–but the others, not so much.

Obviously not homework.

(I put “length” in quotes because I don’t know what the proper term is for what I’m asking about.)

nm

24000 cos(x) where x is your latitude.

The total length of each latitude circle (i.e., its circumference) is proportional to its radius, right? So now you just need to find the radius of each.

Draw a side view of the Earth, sliced into a cross-section. Now the radius of each latitude circle is a line from the axis of the Earth to the surface at that point. Look a bit closer, and you’ll realize you’ve made a right triangle, with the hypotenuse being the radius of the Earth, one of the angles being the latitude, and one of the sides being the radius you’re looking for. Now think back to those trig classes you took back in high school.

Cool. Thanks.

This. I have to do this sort of thing in my job and this is a great method. Not perfect but close enough most of the time.

Note that the distances computed here are those for traveling along the line of latitude. The distance between two points on the same line of latitude and different longitudes bears the same relation to the distance between the two points on the equator with those two latitudes, namely (equator distance) x cos(latitude).

However, the shortest distance between those two points is less (provided they’re not on the equator or one of the poles) as you’d want to travel a great circle route.

The cos(lat)x2400 formula assumes the earth is round, which it is close enough to for almost any purpose. The exact shape is a bit of an oblate spheroid, so the diameter is slightly bigger at the equator than at the poles. The quoted formula will be a bit long for mid latitudes. Lines of latitude are angular measures between the horizon and the pole star, so are evenly spaced by angle, but not by distance along a line of longitude.

For some standards of “evenly spaced by angle”, at least. Your standard of “pole above the horizon” will not give the same results as “angle from the center of the Earth”.

Granted. Latitude was traditionally determined with a sextant, measuring the elevation the pole star above the horizon. In this age of GPS navigation we can now reference the center of the earth, and know and in a few cases even care that the surface is not spherical.

Nitpick: Only the Equator is a (great) circle, all the other lines of latitude are lines constantly curving slightly north or south, and 90°N or S are just points (although from the side they are parallel lines, so your equations are correct…)

Could you elaborate? I’m not quite sure I understand what you mean by “lines constantly curving slightly north or south”. Lines (or more accurately, parallels) of latitude are always parallel to the equator, and while they may appear to curve north or south on a plane projection, that doesn’t change the fact that they are sections of a sphere, and therefore circles.

Yes, but no. They are circles, but not great circles. So, the shortest curve on the surface of the earth between two points on one of those circles will not follow the circle, but will follow a great circle. One property of these great circles is that their centre is the centre of the Earth – that’s not true for these circles of constant latitude.

I’m aware that they aren’t great circles, my confusion was with the assertion that they aren’t circles at all. Am I misunderstanding something in the terminology here?

Chronos made no mention of great circles in his post, to which the above is a response. Unless you’re implying that only great circles are called circles, what you’re saying is true, but irrelevant. The original question was nothing to do with the shortest distance between two points on the surface of the Earth, but specifically about how to calculate the circumference of a latitude circle.

Well, when we talk distance at constant latitude it is pretty natural for great circles to enter the conversation.

It is only in the last couple of centuries that great circles matter much in practice. Prior to the development of the marine chronometer it was impossible to accurately determine longitude. In those days, navigation was by latitude alone, and ships would often sail longer constant latitude routes to avoid getting lost. You could cut a diagonal at the start of the voyage, but needed to lock onto the latitude of you destination long before ( days if not weeks) you expected to arrive there, lest you miss.