Area of a Triangle

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    Area of a Triangle

    Result

    A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    Definition

    Definition

    Defining inequalities

    =0 and y (a^2 + c^2) + x sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))=y (b^2 + c^2)" style="margin-top: 7px;" />

    Lamina properties

    (c, 0) | ((-a^2 + b^2 + c^2)/(2 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)) | (0, 0)

    3

    0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b" style="margin-top: 7px;" />

    (data not available)

    sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)

    x^_ = ((-a^2 + b^2 + 3 c^2)/(6 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(6 c))

    Mechanical properties

    J_x invisible comma x = (-(a - b - c) (a + b - c) (a - b + c) (a + b + c))^(3/2)/(96 c^2)

    J_y invisible comma y = (sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4))/(96 c^2)

    J_zz = -1/48 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (a^2 - 3 (b^2 + c^2))

    J_x invisible comma y = -((a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 - b^2 - 2 c^2))/(96 c^2)

    r_x = ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)/(sqrt(6) c) r_y = sqrt(4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4)/(sqrt(6) c ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4))

    Distance properties

    a | b | c

    p = a + b + c

    r = 1/2 sqrt(-((a - b - c) (a + b - c) (a - b + c))/(a + b + c))

    R = (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    max(a, b, c)

    ? = 1

    s^_ = 2/15 (a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c) (log((a + b + c)/(2 (1/2 (a + b + c) - a)))/a^3 + log((a + b + c)/(2 (1/2 (a + b + c) - b)))/b^3 + log((a + b + c)/(2 (1/2 (a + b + c) - c)))/c^3) + ((b - c)^2 (b + c))/(30 a^2) + ((c - a)^2 (a + c))/(30 b^2) + ((a + b) (a - b)^2)/(30 c^2) + 1/15 (a + b + c)

    A^_ = 1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    Alternate form

    1/4 sqrt((-a - b - c) (a - b - c) (a + b - c) (a - b + c))

    Alternate forms assuming a, b, and c are positive

    1/4 sqrt(a + b - c) sqrt(a - b + c) sqrt(-a + b + c) sqrt(a + b + c) (-1)^(?-(arg(a + b - c) + arg(a - b + c) + arg(-a + b + c) - ?)/(2 ?)?)

    1/4 sqrt(a + b - c) sqrt(a - b + c) sqrt(-a + b + c) sqrt(a + b + c) exp(i ? floor(-arg(a + b - c)/(2 ?) - arg(a - b + c)/(2 ?) - arg(-a + b + c)/(2 ?) + 1/2))

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