Nine-sided 36071

Calculate the surface area and volume of a regular nine-sided pyramid if the radius of the circle inscribed in the base measures ρ = 12 cm and the height of the pyramid is 24 cm

Correct answer:

S =  1526.4708 cm²
V =  3773.6434 cm³

Step-by-step explanation:

$$\begin{gathered} \rho =12\text{cm} \\ h=24\text{cm} \\ n=9 \\ a=2\cdot \rho \cdot \tan (\pi /n)=2\cdot 12\cdot \tan (3.1416/9)\doteq 8.7353\text{cm} \\ {S}_1=n\cdot \frac{a\cdot \rho }{2}=9\cdot \frac{8.7353\cdot 12}{2}\doteq 471.7054{\text{cm}}^2 \\ {h}_2=\sqrt{{\rho }^2+h^2}=\sqrt{12^2+24^2}=12\sqrt{5}\text{cm}\doteq 26.8328\text{cm} \\ {S}_2=\frac{a\cdot h_2}{2}=\frac{8.7353\cdot 26.8328}{2}\doteq 117.1962{\text{cm}}^2 \\ S=S_1+n\cdot S_2=471.7054+9\cdot 117.1962=1526.4708{\text{cm}}^2 \end{gathered}$$

$V=\frac{1}{3}\cdot S_1\cdot h=\frac{1}{3}\cdot 471.7054\cdot 24=3773.6434{\text{cm}}^3$

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