Hexagon cut pyramid

Calculate the volume of a regular 6-sided cut pyramid if the bottom edge is 30 cm, the top edge is 12 cm, and the side edge length is 41 cm.

Correct answer:

V =  44790.6808 cm³

Step-by-step explanation:

$$\begin{gathered} a_1=30\text{cm} \\ {a}_2=12\text{cm} \\ s=41\text{cm} \\ n=6 \\ {S}_1=n\cdot \sqrt{3}/4\cdot a_1^2=6\cdot \sqrt{3}/4\cdot 30^2=1350\sqrt{3}{\text{cm}}^2\doteq 2338.2686{\text{cm}}^2 \\ {S}_2=n\cdot \sqrt{3}/4\cdot a_2^2=6\cdot \sqrt{3}/4\cdot 12^2=216\sqrt{3}{\text{cm}}^2\doteq 374.123{\text{cm}}^2 \\ h=\sqrt{s^2-(a_1-a_2{)}^2}=\sqrt{41^2-(30-12{)}^2}=\sqrt{1357}\text{cm}\doteq 36.8375\text{cm} \\ {h}_2=h\cdot a_2/(a_1-a_2)=36.8375\cdot 12/(30-12)\doteq 24.5583\text{cm} \\ {h}_1=h+h_2=36.8375+24.5583\doteq 61.3958\text{cm} \\ {V}_1=\frac{1}{3}\cdot S_1\cdot h_1=\frac{1}{3}\cdot 2338.2686\cdot 61.3958\doteq 47853.2914{\text{cm}}^3 \\ {V}_2=\frac{1}{3}\cdot S_2\cdot h_2=\frac{1}{3}\cdot 374.123\cdot 24.5583=48\sqrt{4071}{\text{cm}}^3\doteq 3062.6107{\text{cm}}^3 \\ V=V_1-V_2=47853.2914-3062.6107\doteq 44790.6808{\text{cm}}^3 \\ \text{ Verifying Solution: } \\ {V}_3=\frac{h}{3}\cdot (S_1+\sqrt{S_1\cdot S_2}+S_2)=\frac{36.8375}{3}\cdot (2338.2686+\sqrt{2338.2686\cdot 374.123}+374.123)\doteq 44790.6808{\text{cm}}^3 \\ {V}_3=V \end{gathered}$$

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