Even more coni-mising

[Yoruichi-san]

Even more impressed, the High King Math decides to grant you one final reward. He hands you a circle with radius R of some flexible, heat resistant material, which you may cut a slice out of and then roll into a (right circular) cone, with which he will mold you a final cone of gold. How much do you cut?

[bonanova]

Cut a slice having angle PHI so that a fraction f = (2pi-PHI)/2pi of the disc remains.

Make the cone.

Its radius r = f R

Its height h = sqrt (R² - r²) = R sqrt (1 - f²)

Its volume v = (pi/3) h r² = (pi/3) R³ f² sqrt (1 - f²)

f² sqrt (1 - f²) has a maximum of 2 sqrt (3) / 9 when f = sqrt (2/3) and PHI = 1.153 rad = 66.06^o

v_max = 2 pi R³ sqrt (3) / 27 = ~0.403 R³ .