Pentagonal pyramid

The height of a regular pentagonal pyramid is as long as the edge of the base, 20 cm. Calculate the volume and surface area of the pyramid.

Final Answer:

V =  4587.9397 cm³
S =  1902.113 cm²

Step-by-step explanation:

$$\begin{gathered} a=20\text{cm} \\ h=a=20\text{cm} \\ n=5 \\ x=\sqrt{25+10\cdot \sqrt{5}}\doteq 6.8819\text{cm} \\ r=\frac{x}{10}\cdot a=\frac{6.8819}{10}\cdot 20\doteq 13.7638\text{cm} \\ {S}_1=\frac{x}{4}\cdot a^2=\frac{6.8819}{4}\cdot 20^2\doteq 688.191{\text{cm}}^2 \\ V=\frac{1}{3}\cdot S_1\cdot h=\frac{1}{3}\cdot 688.191\cdot 20=4587.9397{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} h_2=\sqrt{r^2+h^2}=\sqrt{13.7638^2+20^2}\doteq 24.2784\text{cm} \\ {S}_2=\frac{a\cdot h_2}{2}=\frac{20\cdot 24.2784}{2}\doteq 242.7844{\text{cm}}^2 \\ S=S_1+n\cdot S_2=688.191+5\cdot 242.7844=1902.113{\text{cm}}^2 \end{gathered}$$

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