Let's use the result of Hartle's Problem 2-5, that the area of a circle of radius r on a sphere of radius a is A=2 \pi a^{2}[1-\cos (r / a)]

Note that r is the length of a curved line; one that you could walk along if you lived on the surface. (a) This result should reduce to the area of the entire sphere, which you learned in junior high, if we take the appropriate limit. Find that limiting value of r and explain it. (b) Using a definition of the Arctic Circle (or Antarctic Circle) as 66.56 degrees north(or south) latitude, and assuming a spherical Earth of radius 6,357 km, calculate the circumference and area; and find their fractional deviations from the flat Earth results.