Regular quadrilateral pyramid

What is the volume of a regular quadrilateral pyramid if its surface is 576 cm² and the base edge is 16 cm?

Correct answer:

V =  1375.9587 cm³

Step-by-step explanation:

$$\begin{gathered} S=576{\text{cm}}^2 \\ a=16\text{ cm } \\ {S}_1=a^2=16^2=256{\text{cm}}^2 \\ {S}_4=(S-S_4)\mathrm{/}4=(576-144)\mathrm{/}4=144{\text{cm}}^2 \\ {S}_4=a{h}_2\mathrm{/}2 \\ {h}_2=2\cdot S_4\mathrm{/}a=2\cdot 144\mathrm{/}16=18\text{ cm } \\ h=\sqrt{h_2^2-(a\mathrm{/}2{)}^2}=\sqrt{18^2-(16\mathrm{/}2{)}^2}=2\sqrt{65}\text{ cm}\doteq 16.1245\text{ cm } \\ V={\displaystyle \frac{1}{3}}\cdot S_1\cdot h={\displaystyle \frac{1}{3}}\cdot 256\cdot 16.1245=1375.9587{\text{cm}}^3 \end{gathered}$$

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