V = 1/3 π a^2 h (for a circular right cone with center at the origin, height h, radius a)

x^2 + y^2<=(a^2 (h - z)^2)/h^2 and 0<=z<=h

(0, 0, h)

1

h

s = sqrt(a^2 + h^2)

S = π a (sqrt(a^2 + h^2) + a)

x^_ = (0, 0, h/4)

I = (1/20 (3 a^2 + 2 h^2) | 0 | 0 0 | 1/20 (3 a^2 + 2 h^2) | 0 0 | 0 | (3 a^2)/10)

max(2 a, sqrt(a^2 + h^2))

χ = 1