Triangular pyramid

What is the volume of a regular triangular pyramid with a side 3 cm long?

Result

V =  3.182 cm3

Solution:

a=3 cm  h1=a2−(a/2)2=32−(3/2)2≐2.5981 cm  h=h12−(h1/3)2=2.59812−(2.5981/3)2≐6 cm≐2.4495 cm  S1=a⋅ h1/2=3⋅ 2.5981/2≐3.8971 cm2  V=13⋅ S1⋅ h=13⋅ 3.8971⋅ 2.4495≐3.182≐3.182 cm3a=3 \ \text{cm} \ \\ \ \\ h_{1}=\sqrt{ a^2-(a/2)^2 }=\sqrt{ 3^2-(3/2)^2 } \doteq 2.5981 \ \text{cm} \ \\ \ \\ h=\sqrt{ h_{1}^2 -(h_{1}/3)^2 }=\sqrt{ 2.5981^2 -(2.5981/3)^2 } \doteq \sqrt{ 6 } \ \text{cm} \doteq 2.4495 \ \text{cm} \ \\ \ \\ S_{1}=a \cdot \ h_{1}/2=3 \cdot \ 2.5981/2 \doteq 3.8971 \ \text{cm}^2 \ \\ \ \\ V=\dfrac{ 1 }{ 3 } \cdot \ S_{1} \cdot \ h=\dfrac{ 1 }{ 3 } \cdot \ 3.8971 \cdot \ 2.4495 \doteq 3.182 \doteq 3.182 \ \text{cm}^3

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