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Cone Volume Formula

Result

V = 1/3 π a^2 h
(volume enclosed by capping the surface below assuming height h, radius a)

Example plots

Equations

x(u, v) = (a (h - u) cos(v))/h
y(u, v) = (a (h - u) sin(v))/h
z(u, v) = u

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

Surface properties

S = π a sqrt(h^2 + a^2)

ds^2 = a^2/h^2 + 1 du^2 + (a^2 (h - u)^2)/h^2 dv^2

dA = a sqrt(((a^2 + h^2) (h - u)^2)/h^4) du dv

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)

K(u, v) = 0

(for a finite single-napped right cone with axis of symmetry along the z-axis, base in the z = 0 plane of radius a, and of height h)