# Hexagon cut pyramid

Calculate the volume of a regular 6-sided cut pyramid if the bottom edge is 30 cm, the top edge us 12 cm, and the side edge length is 41 cm.

Result

V =  4593.916 cm3

#### Solution:

$a_{ 1 } = 30 \ cm \ \\ a_{ 2 } = 12 \ cm \ \\ s = 41 \ cm \ \\ \ \\ S_{ 1 } = 3 \cdot \ \sqrt{ 3 }/2 \cdot \ a_{ 2 }^2 = 3 \cdot \ \sqrt{ 3 }/2 \cdot \ 12^2 = 216 \ \sqrt{ 3 } \ cm^2 \doteq 374.123 \ cm^2 \ \\ S_{ 2 } = 3 \cdot \ \sqrt{ 3 }/2 \cdot \ a_{ 2 }^2 = 3 \cdot \ \sqrt{ 3 }/2 \cdot \ 12^2 = 216 \ \sqrt{ 3 } \ cm^2 \doteq 374.123 \ cm^2 \ \\ \ \\ x = \sqrt{ s^2-(a_{ 1 }-a_{ 2 })^2 } = \sqrt{ 41^2-(30-12)^2 } = \sqrt{ 1357 } \ cm \doteq 36.8375 \ cm \ \\ \ \\ h_{ 2 } = x \cdot \ a_{ 2 }/(a_{ 1 }-a_{ 2 }) = 36.8375 \cdot \ 12/(30-12) \doteq 24.5583 \ cm \ \\ h_{ 1 } = x+h_{ 2 } = 36.8375+24.5583 \doteq 61.3958 \ cm \ \\ \ \\ V_{ 1 } = S_{ 1 } \cdot \ h_{ 1 }/3 = 374.123 \cdot \ 61.3958/3 \doteq 7656.5266 \ cm^3 \ \\ V_{ 2 } = S_{ 2 } \cdot \ h_{ 2 }/3 = 374.123 \cdot \ 24.5583/3 \doteq 3062.6107 \ cm^3 \ \\ V = V_{ 1 }-V_{ 2 } = 7656.5266-3062.6107 \doteq 4593.916 = 4593.916 \ cm^3$

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