Spin off of "Where are You"

What is ( appx. - 4 significant figures ) the radius of a planet from which you could begin at the North Pole, walk 1 mile South. Then 1 mile ( East or West ) completing a full circle, and 1 mile North again to return to the pole?

I figured I'd continue on with my oversight from the other puzzle.

Originally posted by @joe-shmo
What is ( appx. - 4 significant figures ) the radius of a planet from which you could begin at the North Pole, walk 1 mile South. Then 1 mile ( East or West ) completing a full circle, and 1 mile North again to return to the pole?

I figured I'd continue on with my oversight from the other puzzle.

Originally posted by @handyandy
The radius of the circle you walk is one mile (the distance from the pole to the circle).
The circumference of the same circle is one mile.
How is that possible? Are we on the flat surface of a spherical planet?

Originally posted by @joe-shmo
"The radius of the circle you walk is one mile (the distance from the pole to the circle)."

No, the radius of the 1 mile arc you walk in the southerly direction is unknown ( its the radius of the sphere). When you get to that latitude on the sphere you walk west (or east) a full circle that has a circumference of 1 mile.

"Are we on the flat surface of a spherical planet?" We are not, no tricks of wording. Its a math problem.

Originally posted by @js357
Do I understand the poser correctly?

The wording (“THE radius” ) implies that we are walking on the surface of a sphere whose radius, or distance from the center to any point on the surface, is the same at all points on the surface.

You walk for one mile from a pole along any one longitude line, to a point where it intersects a certain latitude line. T ...[text shortened]... one mile in circumference (same crow).

What is the radius of the sphere?

Is this correct?

Originally posted by @joe-shmo
"The radius of the circle you walk is one mile (the distance from the pole to the circle)."

No, the radius of the 1 mile arc you walk in the southerly direction is unknown ( its the radius of the sphere). When you get to that latitude on the sphere you walk west (or east) a full circle that has a circumference of 1 mile.

"Are we on the flat surface of a spherical planet?" We are not, no tricks of wording. Its a math problem.

Originally posted by @handyandy
Do we agree that the radius of a one-mile circle is 0.159 miles on a flat surface?

Originally posted by @joe-shmo
Sure.

Originally posted by @handyandy
And what we're looking for is the radius of the entire planet.

Originally posted by @joe-shmo
Yes sir.

Originally posted by @handyandy
Just an observation: It's a really small planet.

Originally posted by @joe-shmo
Quite small indeed! More like an astriod.

Originally posted by @handyandy
To determine the planet's radius, do we first calculate the relationship between the one-mile
arc from point A to point B and the shorter (0.159 mile) chord linking point A and point B?

Originally posted by @joe-shmo
No, and FYI your figure of a chord length ( 0.159mi) between A and B must be incorrect because you cannot find the chord without knowing the radius in this case ( we do not know the angle subtended by the 1 mile arc - that is to say "explicitly" ) Hint: We can know the angle implicitly as a function of the radius.

Originally posted by @handyandy
I believe 0.159 mile is the radius of a circle having a circumference of one mile, but apparently that's a non-starter.

We need a mathematician.