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 =  200.6434

Solution:

V=120 A=42+3060=852=42.5 ∘ A1=A∘→ rad=A∘⋅ π180 =42.5∘⋅ 3.1415926180 =0.74176=17π/72  tan⁡A=h/(a/2)  h=a/2⋅ tan⁡A V=13a2 h V=16a3 tan⁡A  a=6⋅ V/tan⁡(A1)3=6⋅ 120/tan⁡(0.7418)3≐9.2277  h=a/2⋅ tan⁡(A1)=9.2277/2⋅ tan⁡(0.7418)≐4.2278 sin⁡A1=h:w  w=h/sin⁡(A1)=4.2278/sin⁡(0.7418)≐6.258  S1=a⋅ w/2=9.2277⋅ 6.258/2≐28.8733  S=a2+4⋅ S1=9.22772+4⋅ 28.8733=200.6434

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