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Math Help - Spatial geometry

  1. #1
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    Spatial geometry

    Calculate the volume of the octahedron whose edges are the segments connecting the centers of adjacent faces of a cube, in function on the edge of the cube.
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  2. #2
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    Hello, Apprentice123!

    Calculate the volume of the octahedron whose edges are the segments connecting
    the centers of adjacent faces of a cube, as a function of the edge of the cube.

    Let x = edge of the cube.

    Side view:
    Code:
          : - - - x - - - :
        - *-------*-------*
        : |     * | *     |
        : |   *   |   *   | x
        : | *     |     * |
        x * - - - + - - - * A
        : | *     |     * |
        : |   *   |   *   | x
        : |     * | *     |
        - *-------*-------* C
                  B   x

    From right triangle ABC, the edge of the octahedran is: . AB \:=\:\frac{x}{\sqrt{2}}

    The octahedron is comprised of two pyramids with square bases.
    The square base has side \frac{x}{\sqrt{2}}; its area is: B = \frac{x^2}{2}
    . . and its height is \frac{x}{2}

    The volume of the pyramid is: . \frac{1}{3}Bh   \:=\:\frac{1}{3}\left(\frac{x^2}{2}\right)\left(\f  rac{x}{2}\right) \;=\;\frac{x^3}{12}

    Therefore, the volume of the octahedron is: . 2 \times \frac{x^3}{12} \:=\:\frac{x^3}{6}

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  3. #3
    Super Member
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    Quote Originally Posted by Soroban View Post
    Hello, Apprentice123!


    Let x = edge of the cube.

    Side view:
    Code:
          : - - - x - - - :
        - *-------*-------*
        : |     * | *     |
        : |   *   |   *   | x
        : | *     |     * |
        x * - - - + - - - * A
        : | *     |     * |
        : |   *   |   *   | x
        : |     * | *     |
        - *-------*-------* C
                  B   x

    From right triangle ABC, the edge of the octahedran is: . AB \:=\:\frac{x}{\sqrt{2}}

    The octahedron is comprised of two pyramids with square bases.
    The square base has side \frac{x}{\sqrt{2}}; its area is: B = \frac{x^2}{2}
    . . and its height is \frac{x}{2}

    The volume of the pyramid is: . \frac{1}{3}Bh   \:=\:\frac{1}{3}\left(\frac{x^2}{2}\right)\left(\f  rac{x}{2}\right) \;=\;\frac{x^3}{12}

    Therefore, the volume of the octahedron is: . 2 \times \frac{x^3}{12} \:=\:\frac{x^3}{6}


    Thank you
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