Pentagonal pyramid

Calculate the volume of a regular 5-side (pentaprism) pyramid ABCDEV; if |AB| = 7.7 cm and a plane ABV, ABC has angle 37 degrees.

Result

V =  135.776 cm

Solution:

$a = 7.7 \ \\ u_{ 2 } = 37 \ \\ u_{ 1 } = 360/5/2 = 36 \ \\ \tan (u_{ 1 }) = a/2/v_{ 1 } \ \\ v_{ 1 } = a/2/\tan( u_{ 1 } ^\circ \rightarrow rad) = a/2/\tan( u_{ 1 } ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ ) = 7.7/2/\tan( 36 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ ) = 5.29907 \ \\ S_{ 1 } = a \cdot \ v_{ 1 }/2 = 7.7 \cdot \ 5.2991/2 \doteq 20.4014 \ \\ S_{ 5 } = 5 \cdot \ S_{ 1 } = 5 \cdot \ 20.4014 \doteq 102.0071 \ \\ \ \\ \tan (u_{ 2 }) = v/v_{ 1 } \ \\ v = v_{ 1 } \cdot \ \tan( u_{ 2 } ^\circ \rightarrow rad) = v_{ 1 } \cdot \ \tan( u_{ 2 } ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ ) = 5.299070393814 \cdot \ \tan( 37 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ ) = 3.99314 \ \\ V = S_{ 5 } \cdot \ v / 3 = 102.0071 \cdot \ 3.9931 / 3 \doteq 135.7761 = 135.776 \ \text{ cm }$

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