Quadrilateral pyramid

A regular quadrilateral pyramid has a volume of 24 dm³ and a base edge a = 4 dm.
Calculate:
a/height of the pyramid
b/sidewall height
c/surface of the pyramid

Correct answer:

h =  4.5 dm
h₂ =  4.9244 dm
S =  55.3954 dm²

Step-by-step explanation:

$$\begin{gathered} V=24{\text{dm}}^3 \\ a=4\text{dm} \\ {S}_1=a^2=4^2=16{\text{dm}}^2 \\ V=\frac{1}{3}{S}_{1}h \\ h=\frac{3\cdot V}{S_1}=\frac{3\cdot 24}{16}=\frac{9}{2}\text{dm}=4.5\text{dm} \end{gathered}$$

$h_2=\sqrt{h^2+(a/2{)}^2}=\sqrt{4.5^2+(4/2{)}^2}=4.9244\text{dm}$

$$\begin{gathered} S_2=\frac{a\cdot h_2}{2}=\frac{4\cdot 4.9244}{2}=\sqrt{97}{\text{dm}}^2\doteq 9.8489{\text{dm}}^2 \\ S=S_1+4\cdot S_2=16+4\cdot 9.8489=55.3954{\text{dm}}^2 \end{gathered}$$

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