Volume of what remains of the cuboid

Geometry Level 3

A rectangular cube (cuboid) is cut as shown, so that the cut goes through vertices E E , F F , and G G , and the triangular pyramid is removed. What is volume of what remains of the cuboid, if a = 7 a=7 , b = 5 b=5 , and c = 4 c=4 ?

116.67 93.33 105 102.45

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2 solutions

Let the vertex of the pyramid be H H . The volume of the pyramid E F G H EFGH is given by:

V pyramid = 1 3 × base area × height = 1 3 × [ E F H ] × H G = 1 3 × a b 2 × c = a b c 6 \begin{aligned} V_{\text{pyramid}} & = \frac 13 \times \text{ base area } \times \text{ height} \\ & = \frac 13 \times [EFH] \times \overline{HG} \\ & = \frac 13 \times \frac {ab}2 \times c \\ & = \frac {abc}6 \end{aligned}

Then the volume of what remains is given by:

V remains = V cuboid V pyramid = a b c a b c 6 = 5 6 a b c 116.67 \begin{aligned} V_{\text{remains}} & = V_{\text{cuboid}} - V_{\text{pyramid}} \\ & = abc - \frac {abc}6 \\ & = \frac 56 abc \\ & \approx \boxed{116.67} \end{aligned}

M Zadeh
Jun 3, 2017

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