Pentagonal pyramid

Find the volume and surface of a regular pentagonal pyramid with a base edge a = 12.8 cm and a height v = 32.1 cm.

Correct answer:

V =  3016.1483 cm³
S =  1347.0581 cm²

Step-by-step explanation:

$$\begin{gathered} a=12.8\text{ cm } \\ v=32.1\text{ cm } \\ n=5 \\ {S}_1={\displaystyle \frac{1}{4}}\cdot \sqrt{5\cdot (5+2\cdot \sqrt{5})}\cdot a^2={\displaystyle \frac{1}{4}}\cdot \sqrt{5\cdot (5+2\cdot \sqrt{5})}\cdot 12.8^2\doteq 281.883{\text{cm}}^2 \\ V={\displaystyle \frac{1}{3}}\cdot S_1\cdot v={\displaystyle \frac{1}{3}}\cdot 281.883\cdot 32.1=3016.1483{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} S_1=n\cdot A \\ T=S_1\mathrm{/}n=281.883\mathrm{/}5\doteq 56.3766{\text{cm}}^2 \\ T={\displaystyle \frac{a\cdot h_1}{2}} \\ {h}_1={\displaystyle \frac{2\cdot T}{a}}={\displaystyle \frac{2\cdot 56.3766}{12.8}}\doteq 8.8088\text{ cm } \\ {h}_2^2=v^2+h_1^2 \\ {h}_2=\sqrt{v^2+h_1^2}=\sqrt{32.1^2+8.8088^2}\doteq 33.2867\text{ cm } \\ {S}_2={\displaystyle \frac{a\cdot h_2}{2}}={\displaystyle \frac{12.8\cdot 33.2867}{2}}\doteq 213.035{\text{cm}}^2 \\ S=S_1+n\cdot S_2=281.883+5\cdot 213.035=1347.0581{\text{cm}}^2 \end{gathered}$$

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Showing 2 comments:

Sebf

Perfect!
Thanks for demonstration
Really thanks
Sébastien

4 months ago  Like

Sebf

I think we should see all the variables, I mean : a, v, n S1, S2 and all the things used to finally get the result S. On the scheme.
I understand the way you calculate on your savant literature. But I 'm sure i' m not the only one who would like to learn about this science visually.
For me it's OK. But I cannot share that.
Very sad while it is very clever.
Please explain how you get the result visually on the scheme.

Thanks by advance

Regards

4 months ago  Like

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