Areas and volumes for a sphere | Universal Hyperbolic Geometry 35 | NJ Wildberger

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.

Video Content:
00:00 Introduction
7:16 Archimedes crowning discovery
9:32 Rational turn-angles (tangles)
14:09 Basic tangle formulas
18:38 Proportionality principle
22:39 Relative area of a lune
25:57 Harriot's theorem

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Видео Areas and volumes for a sphere | Universal Hyperbolic Geometry 35 | NJ Wildberger канала Insights into Mathematics