Rectangular Right Prism
abs(x)<=a/2 and abs(y)<=b/2 and abs(z)<=c/2
(for a cuboid with center at the origin and edges along the Cartesian axes of lengths a, b, and c, respectively)
(-a/2, -b/2, -c/2) | (-a/2, -b/2, c/2) | (-a/2, b/2, -c/2) | (-a/2, b/2, c/2) | (a/2, -b/2, -c/2) | (a/2, -b/2, c/2) | (a/2, b/2, -c/2) | (a/2, b/2, c/2)
8
c
S = 2 (a b + a c + b c)
x^_ = (0, 0, 0)
V = a b c
I = (1/12 (b^2 + c^2) | 0 | 0 0 | 1/12 (a^2 + c^2) | 0 0 | 0 | 1/12 (a^2 + b^2))
(for a cuboid with center at the origin and edges along the Cartesian axes of lengths a, b, and c, respectively)
sqrt(a^2 + b^2 + c^2)
χ = 1
s^_ = (90 a^2 b^2 sqrt(a^2 + b^2 + c^2) c^2 + 105 a b (sinh^(-1)(c/sqrt(a^2 + b^2)) b a^3 + sinh^(-1)(b/sqrt(a^2 + c^2)) c a^3 + sinh^(-1)(c/sqrt(a^2 + b^2)) b^3 a + sinh^(-1)(b/sqrt(a^2 + c^2)) c^3 a + sinh^(-1)(a/sqrt(b^2 + c^2)) b c^3 + sinh^(-1)(a/sqrt(b^2 + c^2)) b^3 c) c - 84 a b (tan^(-1)((b c)/(a sqrt(a^2 + b^2 + c^2))) a^4 + tan^(-1)((a c)/(b sqrt(a^2 + b^2 + c^2))) b^4 + tan^(-1)((a b)/(c sqrt(a^2 + b^2 + c^2))) c^4) c + 21 (sinh^(-1)(b/a) b a^6 + sinh^(-1)(c/a) c a^6 + sinh^(-1)(a/b) b^6 a + sinh^(-1)(a/c) c^6 a + sinh^(-1)(b/c) b c^6 + sinh^(-1)(c/b) b^6 c) - 21 (sinh^(-1)(b/sqrt(a^2 + c^2)) b a^6 + sinh^(-1)(c/sqrt(a^2 + b^2)) c a^6 + sinh^(-1)(a/sqrt(b^2 + c^2)) b^6 a + sinh^(-1)(a/sqrt(b^2 + c^2)) c^6 a + sinh^(-1)(b/sqrt(a^2 + c^2)) b c^6 + sinh^(-1)(c/sqrt(a^2 + b^2)) b^6 c) + 25 (b^2 (sqrt(a^2 + b^2) - sqrt(a^2 + b^2 + c^2)) a^4 + c^2 (sqrt(a^2 + c^2) - sqrt(a^2 + b^2 + c^2)) a^4 + b^4 (sqrt(a^2 + b^2) - sqrt(a^2 + b^2 + c^2)) a^2 + c^4 (sqrt(a^2 + c^2) - sqrt(a^2 + b^2 + c^2)) a^2 + b^2 c^4 (sqrt(b^2 + c^2) - sqrt(a^2 + b^2 + c^2)) + b^4 c^2 (sqrt(b^2 + c^2) - sqrt(a^2 + b^2 + c^2))) + 8 ((a - sqrt(a^2 + b^2) - sqrt(a^2 + c^2) + sqrt(a^2 + b^2 + c^2)) a^6 + b^6 (b - sqrt(a^2 + b^2) - sqrt(b^2 + c^2) + sqrt(a^2 + b^2 + c^2)) + c^6 (c - sqrt(a^2 + c^2) - sqrt(b^2 + c^2) + sqrt(a^2 + b^2 + c^2))))/(630 a^2 b^2 c^2)
convex solids | hexahedra | parallelepipeds | solid polyhedra