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Solutions: 4

The angle defining the wedge is in a plane normal to the diameter shared by its sides; let it be a fraction a of a full circle, 0 < a < 1. Then the volume and lune surface area of the wedge are the corresponding fractions of a sphere of radius r:

V = (4/3)rThe lune area 4r^{3}a πS = 4r

^{2}a π + 2 semicircular sides = 4r^{2}a π + r^{2}π

V = SIf r = 3, then the semicircle area divided by π is r(4r - 12)a = 3

Since r

for some integer k. The lune area divided by π is

which is irrational unless r is an integer. We then have

Since 4r

r a Angle Semicircle area Lune area V=S 4 3/4 270° 8 π 48 π 64 π 6 1/4 90° 18 π 36 π 72 π 12 1/12 30° 72 π 48 π 192 π 30 1/36 10° 450 π 100 π 1000 π

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