The beautiful formulas for the surface area and volume of a sphere go back to Archimedes, who also discovered some other remarkable facts relating spheres to circumscribing cylinders. We describe these results.Then we introduce rational turn angles---a renormalization of the notion of angle so that perpendicular lines are represented not by 90 degrees, or by pi/2 radians, but rather by 1/4 turn. This is mathematically the most natural parametrization of an angle, and we restate the sum of angles in a triangle and quadrilateral in terms of turn angles. We state a useful Proportionality Principle.A famous theorem of Harriot (or Girard) gives the ratio of the area of a spherical triangle to the area of the sphere in terms of the sum of turn angles.
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