Tetrahedral pyramid

A regular tetrahedral pyramid is given. Base edge length a = 6.5 cm, side edge s = 7.5 cm. Calculate the volume and the area of its face (side area).

Correct answer:

V =  83.4668 cm³
S =  87.8703 cm²

Step-by-step explanation:

$$\begin{gathered} a=6.5\text{ cm } \\ s=7.5\text{ cm } \\ {S}_1=a^2=6.5^2={\displaystyle \frac{169}{4}}=42.25{\text{cm}}^2 \\ {u}_1=\sqrt{2}\cdot a=\sqrt{2}\cdot 6.5\doteq 9.1924\text{ cm } \\ h=\sqrt{s^2-(u_1\mathrm{/}2{)}^2}=\sqrt{7.5^2-(9.1924\mathrm{/}2{)}^2}\doteq 5.9266\text{ cm } \\ V={\displaystyle \frac{1}{3}}\cdot S_1\cdot h={\displaystyle \frac{1}{3}}\cdot 42.25\cdot 5.9266=83.4668{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} h_2=\sqrt{h^2+(a\mathrm{/}2{)}^2}=\sqrt{5.9266^2+(6.5\mathrm{/}2{)}^2}\doteq 6.7593\text{ cm } \\ {S}_2={\displaystyle \frac{1}{2}}\cdot a\cdot h_2={\displaystyle \frac{1}{2}}\cdot 6.5\cdot 6.7593\doteq 21.9676{\text{cm}}^2 \\ S=4\cdot S_2=4\cdot 21.9676=87.8703{\text{cm}}^2 \end{gathered}$$

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