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    Polyhedron Centroid

    Result

    cube | x^_ = (0, 0, 0) cuboid | x^_ = (0, 0, 0) general tetrahedron | x^_ = ((4 d_12^2 + d_13^2 + d_14^2 - d_23^2 - d_24^2)/(8 d_12), -(2 d_12^4 - 3 d_12^2 d_13^2 + d_13^4 - d_12^2 d_14^2 + d_13^2 d_14^2 - 3 d_12^2 d_23^2 - 2 d_13^2 d_23^2 - d_14^2 d_23^2 + d_23^4 - d_12^2 d_24^2 - d_13^2 d_24^2 + d_23^2 d_24^2 + 2 d_12^2 d_34^2)/(8 d_12 sqrt((d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (-d_12 + d_13 + d_23) (d_12 + d_13 + d_23))), -(d_12^2 d_13^2 d_23^2 - d_12^2 d_14^2 d_23^2 - d_13^2 d_14^2 d_23^2 + d_14^4 d_23^2 + d_14^2 d_23^4 - d_12^2 d_13^2 d_24^2 + d_13^4 d_24^2 + d_12^2 d_14^2 d_24^2 - d_13^2 d_14^2 d_24^2 - d_13^2 d_23^2 d_24^2 - d_14^2 d_23^2 d_24^2 + d_13^2 d_24^4 + d_12^4 d_34^2 - d_12^2 d_13^2 d_34^2 - d_12^2 d_14^2 d_34^2 + d_13^2 d_14^2 d_34^2 - d_12^2 d_23^2 d_34^2 - d_14^2 d_23^2 d_34^2 - d_12^2 d_24^2 d_34^2 - d_13^2 d_24^2 d_34^2 + d_23^2 d_24^2 d_34^2 + d_12^2 d_34^4)/(2 sqrt(2) sqrt((d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (-d_12 + d_13 + d_23) (d_12 + d_13 + d_23)) sqrt( left bracketing bar 0 | 1 | 1 | 1 | 1 1 | 0 | d_12^2 | d_13^2 | d_14^2 1 | d_12^2 | 0 | d_23^2 | d_24^2 1 | d_13^2 | d_23^2 | 0 | d_34^2 1 | d_14^2 | d_24^2 | d_34^2 | 0 right bracketing bar ))) isosceles tetrahedron | x^_ = (a/2, (a (-a^2 + b^2 + c^2))/(2 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))), sqrt(((a^2 + b^2 - c^2) (a^2 - b^2 + c^2) (-a^2 + b^2 + c^2))/((a + b - c) (a - b + c) (-a + b + c) (a + b + c)))/(2 sqrt(2))) regular tetrahedron | x^_ = (0, 0, 0) trirectangular tetrahedron | x^_ = (a/4, b/4, c/4) square pyramid | x^_ = (0, 0, h/4)

    Plots

    Plots Cube

    Plots Cuboid

    Equation

    2 z<=a and 2 x<=a and 2 y<=a and 2 x>=-a and 2 z>=-a and 2 y>=-a

    abs(x)<=a/2 and abs(y)<=b/2 and abs(z)<=c/2

    sqrt(-(d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23)) z>=0 and d_12 ((sqrt(d_34^2 d_12^4 + (d_34^4 - (d_13^2 + d_14^2 + d_23^2 + d_24^2) d_34^2 + (d_13 - d_14) (d_13 + d_14) (d_23 - d_24) (d_23 + d_24)) d_12^2 + d_13^4 d_24^2 + d_23^2 (d_14^4 + (d_23^2 - d_24^2 - d_34^2) d_14^2 + d_24^2 d_34^2) - d_13^2 ((d_23^2 + d_24^2 - d_34^2) d_14^2 + d_24^2 (d_23^2 - d_24^2 + d_34^2))) y)/sqrt((d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23)) + ((d_12^4 - (d_13^2 + d_14^2 + d_23^2 + d_24^2 - 2 d_34^2) d_12^2 + (d_13^2 - d_23^2) (d_14^2 - d_24^2)) z)/(2 d_12 sqrt(-(d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23))))>=0 and (1/2 sqrt(-(d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23)) ((z d_14^2)/d_12 - (z d_24^2)/d_12 + z d_12 - (2 x sqrt(d_34^2 d_12^4 + (d_34^4 - (d_13^2 + d_14^2 + d_23^2 + d_24^2) d_34^2 + (d_13 - d_14) (d_13 + d_14) (d_23 - d_24) (d_23 + d_24)) d_12^2 + d_13^4 d_24^2 + d_23^2 (d_14^4 + (d_23^2 - d_24^2 - d_34^2) d_14^2 + d_24^2 d_34^2) - d_13^2 ((d_23^2 + d_24^2 - d_34^2) d_14^2 + d_24^2 (d_23^2 - d_24^2 + d_34^2))))/sqrt((d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23))) + (d_12^2 + d_13^2 - d_23^2) ((sqrt(d_34^2 d_12^4 + (d_34^4 - (d_13^2 + d_14^2 + d_23^2 + d_24^2) d_34^2 + (d_13 - d_14) (d_13 + d_14) (d_23 - d_24) (d_23 + d_24)) d_12^2 + d_13^4 d_24^2 + d_23^2 (d_14^4 + (d_23^2 - d_24^2 - d_34^2) d_14^2 + d_24^2 d_34^2) - d_13^2 ((d_23^2 + d_24^2 - d_34^2) d_14^2 + d_24^2 (d_23^2 - d_24^2 + d_34^2))) y)/sqrt((d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23)) + (z (d_12^4 - (d_13^2 + d_14^2 + d_23^2 + d_24^2 - 2 d_34^2) d_12^2 + (d_13^2 - d_23^2) (d_14^2 - d_24^2)))/(2 d_12 sqrt(-(d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23)))))/(2 d_12)<=0 and 1/2 ((sqrt(d_34^2 d_12^4 + (d_34^4 - (d_13^2 + d_14^2 + d_23^2 + d_24^2) d_34^2 + (d_13 - d_14) (d_13 + d_14) (d_23 - d_24) (d_23 + d_24)) d_12^2 + d_13^4 d_24^2 + d_23^2 (d_14^4 + (d_23^2 - d_24^2 - d_34^2) d_14^2 + d_24^2 d_34^2) - d_13^2 ((d_23^2 + d_24^2 - d_34^2) d_14^2 + d_24^2 (d_23^2 - d_24^2 + d_34^2))) (y d_12^2 - sqrt(-(d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23)) d_12 - y d_13^2 + y d_23^2 + x sqrt(-(d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23))))/(d_12 sqrt((d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23))) + (z ((d_23^2 - d_24^2 + d_34^2) d_12^2 + d_13^2 (d_23^2 + d_24^2 - d_34^2) + d_23^2 (-2 d_14^2 - d_23^2 + d_24^2 + d_34^2)))/sqrt(-(d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23)))<=0

    z>=0 and 2 sqrt(2) a y sqrt((a^2 + b^2 - c^2) (a^2 - b^2 + c^2) (-a^2 + b^2 + c^2)) - z (-3 a^4 + 2 a^2 b^2 + 2 a^2 c^2 + b^4 - 2 b^2 c^2 + c^4)>=0 and sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) (z sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) (a^2 + b^2 - c^2) - 2 sqrt(2) a x sqrt((a^2 + b^2 - c^2) (a^2 - b^2 + c^2) (-a^2 + b^2 + c^2))) + (a^2 - b^2 + c^2) (2 sqrt(2) a y sqrt((a^2 + b^2 - c^2) (a^2 - b^2 + c^2) (-a^2 + b^2 + c^2)) - z (-3 a^4 + 2 a^2 b^2 + 2 a^2 c^2 + b^4 - 2 b^2 c^2 + c^4))<=0 and sqrt(2) sqrt((a^2 + b^2 - c^2) (a^2 - b^2 + c^2) (-a^2 + b^2 + c^2)) (y (a^2 + b^2 - c^2) + (x - a) sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))) + a z (a^4 + 2 a^2 b^2 - 2 a^2 c^2 - 3 b^4 + 2 b^2 c^2 + c^4)<=0

    sqrt(2) a + 4 sqrt(3) z>=0 and 4 sqrt(3) z<=3 sqrt(2) a + 8 sqrt(6) x and 4 (sqrt(6) x - 3 sqrt(2) y + sqrt(3) z)<=3 sqrt(2) a and 4 (sqrt(6) x + 3 sqrt(2) y + sqrt(3) z)<=3 sqrt(2) a

    z>=0 and y>=0 and x>=0 and a b z + a c y + b c x<=a b c

    0<=z<=h and max(abs(x), abs(y))<=1/2 a (1 - z/h)

    Solid properties

    | vertices cube | (-a/2, -a/2, -a/2) | (-a/2, -a/2, a/2) | (-a/2, a/2, -a/2) | (-a/2, a/2, a/2) | (a/2, -a/2, -a/2) | (a/2, -a/2, a/2) | (a/2, a/2, -a/2) | (a/2, a/2, a/2) cuboid | (-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) general tetrahedron | (0, 0, 0) | (d_12, 0, 0) | ((d_12^2 + d_13^2 - d_23^2)/(2 d_12), sqrt(-(d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23))/(2 d_12), 0) | ((d_12^2 + d_14^2 - d_24^2)/(2 d_12), (-d_12^4 - (d_13 - d_23) (d_13 + d_23) (d_14 - d_24) (d_14 + d_24) + d_12^2 (d_13^2 + d_14^2 + d_23^2 + d_24^2 - 2 d_34^2))/(2 d_12 sqrt(-(d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23))), sqrt(d_13^4 d_24^2 + d_12^4 d_34^2 + d_12^2 ((d_13 - d_14) (d_13 + d_14) (d_23 - d_24) (d_23 + d_24) - (d_13^2 + d_14^2 + d_23^2 + d_24^2) d_34^2 + d_34^4) + d_23^2 (d_14^4 + d_24^2 d_34^2 + d_14^2 (d_23^2 - d_24^2 - d_34^2)) - d_13^2 (d_14^2 (d_23^2 + d_24^2 - d_34^2) + d_24^2 (d_23^2 - d_24^2 + d_34^2)))/sqrt((d_12 - d_13 - d_23) (d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (d_12 + d_13 + d_23))) isosceles tetrahedron | (0, 0, 0) | (a, 0, 0) | ((a^2 - b^2 + c^2)/(2 a), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 a), 0) | ((a^2 + b^2 - c^2)/(2 a), (-3 a^4 + (b^2 - c^2)^2 + 2 a^2 (b^2 + c^2))/(2 a sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))), sqrt(2) sqrt(((a^2 + b^2 - c^2) (a^2 - b^2 + c^2) (-a^2 + b^2 + c^2))/((a + b - c) (a - b + c) (-a + b + c) (a + b + c)))) regular tetrahedron | (0, 0, (sqrt(2/3) - 1/(2 sqrt(6))) a) | (-a/(2 sqrt(3)), -a/2, -a/(2 sqrt(6))) | (-a/(2 sqrt(3)), a/2, -a/(2 sqrt(6))) | (a/sqrt(3), 0, -a/(2 sqrt(6))) trirectangular tetrahedron | (0, 0, 0) | (0, 0, c) | (0, b, 0) | (a, 0, 0) square pyramid | (-a/2, -a/2, 0) | (a/2, -a/2, 0) | (a/2, a/2, 0) | (-a/2, a/2, 0) | (0, 0, h) | number of vertices cube | 8 cuboid | 8 general tetrahedron | 4 isosceles tetrahedron | 4 regular tetrahedron | 4 trirectangular tetrahedron | 4 | height cuboid | c isosceles tetrahedron | sqrt(2) sqrt(((a^2 + b^2 - c^2) (a^2 - b^2 + c^2) (-a^2 + b^2 + c^2))/((a + b - c) (a - b + c) (-a + b + c) (a + b + c))) regular tetrahedron | sqrt(2/3) a trirectangular tetrahedron | c square pyramid | h | slant height square pyramid | s = sqrt(a^2/4 + h^2) | surface area cube | S = 6 a^2 cuboid | S = 2 (a b + a c + b c) general tetrahedron | S = 1/4 (sqrt((d_12 + d_13 - d_23) (d_12 - d_13 + d_23) (-d_12 + d_13 + d_23) (d_12 + d_13 + d_23)) + sqrt((d_12 + d_14 - d_24) (d_12 - d_14 + d_24) (-d_12 + d_14 + d_24) (d_12 + d_14 + d_24)) + sqrt((d_13 + d_14 - d_34) (d_13 - d_14 + d_34) (-d_13 + d_14 + d_34) (d_13 + d_14 + d_34)) + sqrt((d_23 + d_24 - d_34) (d_23 - d_24 + d_34) (-d_23 + d_24 + d_34) (d_23 + d_24 + d_34))) isosceles tetrahedron | S = sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) regular tetrahedron | S = sqrt(3) a^2 trirectangular tetrahedron | S = 1/2 (sqrt(a^2 b^2 + a^2 c^2 + b^2 c^2) + a b + a c + b c) square pyramid | S = a sqrt(a^2 + 4 h^2) + a^2 | volume cube | V = a^3 cuboid | V = a b c general tetrahedron | V = sqrt(1/288 left bracketing bar 0 | 1 | 1 | 1 | 1 1 | 0 | d_12^2 | d_13^2 | d_14^2 1 | d_12^2 | 0 | d_23^2 | d_24^2 1 | d_13^2 | d_23^2 | 0 | d_34^2 1 | d_14^2 | d_24^2 | d_34^2 | 0 right bracketing bar ) isosceles tetrahedron | V = sqrt(1/72 (b^2 + c^2 - a^2) (a^2 + c^2 - b^2) (a^2 + b^2 - c^2)) regular tetrahedron | V = a^3/(6 sqrt(2)) trirectangular tetrahedron | V = (a b c)/6 square pyramid | V = (a^2 h)/3 | moment of inertia tensor cube | I = (a^2/6 | 0 | 0 0 | a^2/6 | 0 0 | 0 | a^2/6) cuboid | 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)) isosceles tetrahedron | I = ((sqrt((a - b - c) (a + b - c) (a - b + c) (a + b + c)) (5 a^4 - 6 a^2 (b^2 + c^2) + (b^2 - c^2)^2) sqrt(1/(a^4 - 2 a^2 (b^2 + c^2) + (b^2 - c^2)^2)))/(40 a^2) | (10 a^6 - a^4 (9 b^2 + 11 c^2) - (b^2 - c^2)^3)/(40 a^2 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c))) | ((5 a^2 + b^2 - c^2) sqrt((a - b - c) (a^2 - b^2 - c^2) (a^2 + b^2 - c^2) (a^2 - b^2 + c^2)))/(20 sqrt(2) a (a - b - c) sqrt((a + b - c) (a - b + c) (a + b + c))) (10 a^6 - a^4 (9 b^2 + 11 c^2) - (b^2 - c^2)^3)/(40 a^2 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c))) | (19 a^8 - 30 a^6 (b^2 + c^2) + 4 a^4 (b^2 - c^2)^2 + 6 a^2 (b^2 - c^2)^2 (b^2 + c^2) + (b^2 - c^2)^4)/(40 a^2 (a^4 - 2 a^2 (b^2 + c^2) + (b^2 - c^2)^2)) | (sqrt(-(a^2 - b^2 - c^2) (a^2 + b^2 - c^2) (a^2 - b^2 + c^2)) (-7 a^4 + 6 a^2 (b^2 + c^2) + (b^2 - c^2)^2))/(20 sqrt(2) a (a - b - c) (a + b - c) (a - b + c) (a + b + c)) ((5 a^2 + b^2 - c^2) sqrt((a - b - c) (a^2 - b^2 - c^2) (a^2 + b^2 - c^2) (a^2 - b^2 + c^2)))/(20 sqrt(2) a (a - b - c) sqrt((a + b - c) (a - b + c) (a + b + c))) | (sqrt(-(a^2 - b^2 - c^2) (a^2 + b^2 - c^2) (a^2 - b^2 + c^2)) (-7 a^4 + 6 a^2 (b^2 + c^2) + (b^2 - c^2)^2))/(20 sqrt(2) a (a - b - c) (a + b - c) (a - b + c) (a + b + c)) | -(a^6 - a^4 (b^2 + c^2) - a^2 (b^4 - 26 b^2 c^2 + c^4) + (b^2 - c^2)^2 (b^2 + c^2))/(20 (a^4 - 2 a^2 (b^2 + c^2) + (b^2 - c^2)^2))) regular tetrahedron | I = (a^2/20 | 0 | 0 0 | a^2/20 | 0 0 | 0 | a^2/20) trirectangular tetrahedron | I = (1/10 (b^2 + c^2) | -(a b)/20 | -(a c)/20 -(a b)/20 | 1/10 (a^2 + c^2) | -(b c)/20 -(a c)/20 | -(b c)/20 | 1/10 (a^2 + b^2)) square pyramid | I = (1/20 (a^2 + 2 h^2) | 0 | 0 0 | 1/20 (a^2 + 2 h^2) | 0 0 | 0 | a^2/10)

    Distance properties

    | generalized diameter cube | sqrt(3) a cuboid | sqrt(a^2 + b^2 + c^2) general tetrahedron | max(d_12, d_13, d_14, d_23, d_24, d_34) isosceles tetrahedron | max(a, b, c) regular tetrahedron | a trirectangular tetrahedron | max(a^2 + b^2, a^2 + c^2, b^2 + c^2) square pyramid | max(sqrt(2) a, sqrt(a^2/2 + h^2)) | convexity coefficient cube | χ = 1 cuboid | χ = 1 general tetrahedron | χ = 1 isosceles tetrahedron | χ = 1 regular tetrahedron | χ = 1 trirectangular tetrahedron | χ = 1 | mean line segment length cube | s^_ = a Δ(3) cuboid | 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) | mean tetrahedron volume cube | V^_ = (3977/216000 - π^2/2160) a^3 general tetrahedron | V^_ = sqrt(1/288 left bracketing bar 0 | 1 | 1 | 1 | 1 1 | 0 | d_12^2 | d_13^2 | d_14^2 1 | d_12^2 | 0 | d_23^2 | d_24^2 1 | d_13^2 | d_23^2 | 0 | d_34^2 1 | d_14^2 | d_24^2 | d_34^2 | 0 right bracketing bar ) (13/720 - π^2/15015) isosceles tetrahedron | V^_ = sqrt(1/72 (b^2 + c^2 - a^2) (a^2 + c^2 - b^2) (a^2 + b^2 - c^2)) (13/720 - π^2/15015) regular tetrahedron | V^_ = (a^3 (13/720 - π^2/15015))/(6 sqrt(2)) trirectangular tetrahedron | V^_ = 1/6 a b c (13/720 - π^2/15015)

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