Triangular pyramid

Determine the volume and surface area of a regular triangular pyramid having a base edge a=20 cm and a lateral edge b = 35 cm

Correct answer:

V =  1848.4228 cm³
S =  1179.4357 cm²

Step-by-step explanation:

$$\begin{gathered} a=20\text{ cm } \\ b=35\text{ cm } \\ {a}_2=a\mathrm{/}2=20\mathrm{/}2=10\text{ cm } \\ {h}_1=\sqrt{b^2-a_2^2}=\sqrt{35^2-10^2}=15\sqrt{5}\text{ cm}\doteq 33.541\text{ cm } \\ h=\sqrt{h_1^2-a_2^2}=\sqrt{33.541^2-10^2}=5\sqrt{41}\text{ cm}\doteq 32.0156\text{ cm } \\ {S}_1=a\cdot h_1\mathrm{/}2=20\cdot 33.541\mathrm{/}2=150\sqrt{5}{\text{cm}}^2\doteq 335.4102{\text{cm}}^2 \\ {S}_2={\displaystyle \frac{\sqrt{3}}{4}}\cdot a^2={\displaystyle \frac{\sqrt{3}}{4}}\cdot 20^2=100\sqrt{3}{\text{cm}}^2\doteq 173.2051{\text{cm}}^2 \\ V={\displaystyle \frac{1}{3}}\cdot S_2\cdot h={\displaystyle \frac{1}{3}}\cdot 173.2051\cdot 32.0156=1848.4228{\text{cm}}^3 \end{gathered}$$

$S=3\cdot S_1+S_2=3\cdot 335.4102+173.2051=1179.4357{\text{cm}}^2$

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