# Tetrahedral pyramid

Determine the surface of a regular tetrahedral pyramid when its volume is V = 120 and the angle of the sidewall with the base plane is α = 42° 30´.

Correct result:

S =  200.643

#### Solution:

$V=120 \ \\ A=42 + \dfrac{ 30 }{ 60 }=\dfrac{ 85 }{ 2 }=42.5 \ ^\circ \ \\ A_{1}=A ^\circ \rightarrow\ \text{rad}=A ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ =42.5 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ =0.74176=17π/72 \ \\ \ \\ \tan A=h /(a/2) \ \\ \ \\ h=a/2 \cdot \ \tan A \ \\ V=\dfrac{ 1 }{ 3 } a^2 \ h \ \\ V=\dfrac{ 1 }{ 6 } a^3 \ \tan A \ \\ \ \\ a=\sqrt[3]{ 6 \cdot \ V / \tan(A_{1}) }=\sqrt[3]{ 6 \cdot \ 120 / \tan(0.7418) } \doteq 9.2277 \ \\ \ \\ h=a/2 \cdot \ \tan(A_{1})=9.2277/2 \cdot \ \tan(0.7418) \doteq 4.2278 \ \\ \sin A_{1}=h : w \ \\ \ \\ w=h / \sin(A_{1})=4.2278 / \sin(0.7418) \doteq 6.258 \ \\ \ \\ S_{1}=a \cdot \ w/2=9.2277 \cdot \ 6.258/2 \doteq 28.8733 \ \\ \ \\ S=a^2+4 \cdot \ S_{1}=9.2277^2+4 \cdot \ 28.8733=200.643$

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