Tetrahedral pyramid

Calculate the surface S and the volume V of a regular tetrahedral pyramid with the base side a = 5 m and a body height of 14 m.

Correct answer:

V =  116.6667 m³
S =  167.2146 m²

Step-by-step explanation:

$$\begin{gathered} a=5\text{ m } \\ h=14\text{ m } \\ {S}_1=a^2=5^2=25{\text{m}}^2 \\ V={\displaystyle \frac{1}{3}}\cdot S_1\cdot h={\displaystyle \frac{1}{3}}\cdot 25\cdot 14={\displaystyle \frac{350}{3}}=116.6667{\text{m}}^3 \end{gathered}$$

$$\begin{gathered} h_2=\sqrt{h^2+(a\mathrm{/}2{)}^2}=\sqrt{14^2+(5\mathrm{/}2{)}^2}\doteq 14.2215\text{ m } \\ {S}_2={\displaystyle \frac{a\cdot h_2}{2}}={\displaystyle \frac{5\cdot 14.2215}{2}}\doteq 35.5537{\text{m}}^2 \\ S=S_1+4\cdot S_2=25+4\cdot 35.5537=167.2146{\text{m}}^2 \end{gathered}$$

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