Quadrangular pyramid

Given is a regular quadrangular pyramid with a square base. The body height is 30 cm and volume V = 1000 cm³. Calculate its side a and its surface area.

Correct result:

a =  10 cm
S =  708.2763 cm²

Solution:

$$\begin{gathered} h=30\text{ cm } \\ V=1000{\text{cm}}^3 \\ V={\displaystyle \frac{1}{3}}h{S}_1 \\ {S}_1=3\cdot V\mathrm{/}h=3\cdot 1000\mathrm{/}30=100{\text{cm}}^2 \\ {S}_1=a^2 \\ a=\sqrt{S_1}=\sqrt{100}=10\text{ cm} \end{gathered}$$

$$\begin{gathered} h_2=\sqrt{h^2+(a\mathrm{/}2{)}^2}=\sqrt{30^2+(10\mathrm{/}2{)}^2}=5\sqrt{37}\text{ cm}\doteq 30.4138\text{ cm } \\ {S}_2=a\cdot h_2\mathrm{/}2=10\cdot 30.4138\mathrm{/}2=25\sqrt{37}{\text{cm}}^2\doteq 152.0691{\text{cm}}^2 \\ S=S_1+4\cdot S_2=100+4\cdot 152.0691=708.2763{\text{cm}}^2 \end{gathered}$$

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