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Putnam 1998 B3

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October 28, 2012
Putnamgeometrytrigonometryprobabilityrotation3D geometrysphere

Problem Statement

Let HH be the unit hemisphere {(x,y,z):x2+y2+z2=1,z≥0}\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}, CC the unit circle {(x,y,0):x2+y2=1}\{(x,y,0):x^2+y^2=1\}, and PP the regular pentagon inscribed in CC. Determine the surface area of that portion of HH lying over the planar region inside PP, and write your answer in the form Asin⁡α+Bcos⁡βA \sin\alpha + B \cos\beta, where A,B,α,βA,B,\alpha,\beta are real numbers.
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