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    Isosceles Tetrahedron

    Example plots

    Example plots

    Equation

    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

    Solid properties

    (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))))

    4

    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)))

    S = sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    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)))

    V = sqrt(1/72 (b^2 + c^2 - a^2) (a^2 + c^2 - b^2) (a^2 + b^2 - c^2))

    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)))

    Distance properties

    max(a, b, c)

    χ = 1

    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)

    (where lengths and volumes refer to line segments and tetrahedra picked at random from points contained in the solid)

    Properties

    convex solids | solid polyhedra | tetrahedra

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