Quadrilateral 82146

Calculate the volume and surface area of a regular quadrilateral prism with base edge a=24 cm if the body diagonal makes an angle of 66° with the base.

Correct answer:

V =  43910.22 cm³
S =  8470.37 cm²

Step-by-step explanation:

$$\begin{gathered} a=24\text{cm} \\ \alpha =66{}^{\circ } \\ {S}_1=a^2=24^2=576{\text{cm}}^2 \\ s=\sqrt{2}\cdot a=\sqrt{2}\cdot 24=24\sqrt{2}\text{cm}\doteq 33.9411\text{cm} \\ \tan \alpha =h:s \\ h=s\cdot \tan \alpha =s\cdot \tan 66°=33.9411\cdot \tan 66°=33.9411\cdot 2.246037=76.23302\text{cm} \\ V=S_1\cdot h=576\cdot 76.233=43910.22{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} o=4\cdot a=4\cdot 24=96\text{cm} \\ {S}_2=o\cdot h=96\cdot 76.233\doteq 7318.3695{\text{cm}}^2 \\ S=2\cdot S_1+S_2=2\cdot 576+7318.3695=8470.37{\text{cm}}^2 \end{gathered}$$

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