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3.3 Sphere 3.3.1 Auxiliary Principle
The plane P intersects the center of a sphere with the radius of R. The plane Q is parallel to P with the distance of h. Find the intersection area of Q in terms of R and r.
ݎ ଶ ൌܵ
ܴ ଶ െ ݄ ଶ ൌ ݎߨ ଶ ቅ ֜ ܵ ൌ ߨ ൈ ሺܴ ଶ െ ݄ ଶ ሻ
3.3.2 Hemisphere
The image above illustrates a cylinder containing a cone and a hemisphere with equal radii. The intersecting surface of each plane of the hemisphere equals the excluding part of the cylinder.
ܴ ൌ ܴ ௬ௗ
െ ݕ
For a random plane passing through the hemisphere, let
Then, the auxiliary principle is taken into account :
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