Suppose you were an alien living on the fictitious warlike planet myrmidon and you wanted to measure its size. the sun is shining directly down a missile silo 1000 miles to your south, while at your location, the sun is 36° from straight overhead. what is the circumference of myrmidon? what is its radius?

The angle of the sun on the equator with the vertical is also the latitude where the sun is exactly overhead.
Similarly, when stationed at any point, the angle of the sun with the vertical is the arc angle with the location where the sun is exactly overhead.  This is also the method used in ancient times in Egypt to measure the size of the earth.

36-degree arc = 1000 miles = r(theta)
r=radius
theta = 36 degrees in radians = 36*pi/180=0.6283
r=1000 miles /0.6283=1591.55 miles
circumference c=2*pi*r=10,000 miles.