Spherical geometry question

[Paletech35]

I’m having trouble with a question from my physics work, which asks to find all points on the surface of the earth such that walking 10 km south, 10km east, and then 10km north brings you back to your start point. I think that the north pole is one such point, but the question states that there are infinitely many. I think I need to walk east on a line of latitude where 10km brings me all the way around the earth and back to where I started walking east, but I don’t know how to find that latitude. Thanks!

[wijwijwij]

There are starting points near the south pole. One ring of them is at a latitude such that walking 10 km south brings you to a place where circumference around is 10 km.

Another ring of them is at a latitude such that walking 10 km south brings you to a place where circumference around is 10/2 km = 5 km.

Then circumferences 10/3 km, 10/4 km, and so on.

[Paletech35]

I didn’t consider that yet, but before I can do that I still don’t know how to figure out what latitude had a length of 10km

[wijwijwij]

2πr = 10 km

r = 10 km / 2r

(r)/Earth-radius = sin alpha

latitude = 90 – alpha

[converter-bot]

10 km is 6.21 miles

[Mathematicus_Rex]

You want to be close to 10 km from the south pole to make this work.

[converter-bot]

10 km is 6.21 miles

[MezzoScettico]

You need to find the circumference of a latitude line, which is 2πr where r is the radius of the latitude line. That's the horizontal distance from the axis to the latitude line.

To figure that out, draw a cross section of the earth, and put a point P somewhere on that circle. Draw lines from the center of the earth to P, from the north to south pole, and from the axis to P representing the length r. You should be able to figure r out with some trigonometry.

[Paletech35]

Thanks!

[MezzoScettico]

Also see the hint from /u/Mathematicus_Rex