the largest circle (spherical cap) drawable on a sphere of radius r is the whole sphere (perhaps minus one point). the circle has radius pi*r (measured on the surface of the sphere) and area 4*pi*r^2 . (incidentally, the perimeter, the length of its outer edge, is zero or infinitesimal.) a flat circle with that radius has area pi*(pi*r)^2 = pi^3*r^2. ratio of curved versus flat area = 4/pi^2 ~= 0.405 .
hemisphere is also a circle. curved area = 2*pi*r^2. radius = pi*r/2. flat area with that radius = pi^3*r^2/4. area ratio = 8/pi^2 = 0.811 .
general spherical cap: curved area = 2 * pi * r^2 * (1-cos(theta)). radius = theta * r. flat area = pi * theta^2 * r^2. ratio = 2 * (1-cos(theta))/theta^2. (future work: show that this ratio approaches 1 in the limit as theta approaches zero, that is, when a spherical cap is small, it behaves as if it were flat. this should be easy.)
previously: packing many circles onto the surface of a sphere.
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