Glass

How many glasses are needed to produce glass with a base of a regular 5-gon if one base triangle in the base is 4.2 square cm and the height is 10 cm?

Correct answer:

S =  195.6848 cm²

Step-by-step explanation:

$$\begin{gathered} n=5 \\ {S}_1=n\cdot 4.2=5\cdot 4.2=21{\text{cm}}^2 \\ h=10\text{cm} \\ \Phi =\frac{360}{n}=\frac{360}{5}=72{}^{\circ } \\ \alpha =\frac{180-\Phi }{2}=\frac{180-72}{2}=54{}^{\circ } \\ {S}_1=n/4a^2\tan \alpha \\ a=\sqrt{\frac{4\cdot S_1}{n\cdot \tan \alpha }}=\sqrt{\frac{4\cdot S_1}{n\cdot \tan 54°}}=\sqrt{\frac{4\cdot 21}{5\cdot \tan 54°}}=\sqrt{\frac{4\cdot 21}{5\cdot 1.376382}}=3.4937\text{cm} \\ o=n\cdot a=5\cdot 3.4937\doteq 17.4685\text{cm} \\ S=S_1+o\cdot h=21+17.4685\cdot 10=195.6848{\text{cm}}^2 \end{gathered}$$

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