# Triangular pyramid

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

Result

V =  3.182 cm3

#### Solution:

$a=3 \ \text{cm} \ \\ \ \\ h_{1}=\sqrt{ a^2-(a/2)^2 }=\sqrt{ 3^2-(3/2)^2 } \doteq 2.5981 \ \text{cm} \ \\ \ \\ h=\sqrt{ h_{1}^2 -(h_{1}/3)^2 }=\sqrt{ 2.5981^2 -(2.5981/3)^2 } \doteq \sqrt{ 6 } \ \text{cm} \doteq 2.4495 \ \text{cm} \ \\ \ \\ S_{1}=a \cdot \ h_{1}/2=3 \cdot \ 2.5981/2 \doteq 3.8971 \ \text{cm}^2 \ \\ \ \\ V=\dfrac{ 1 }{ 3 } \cdot \ S_{1} \cdot \ h=\dfrac{ 1 }{ 3 } \cdot \ 3.8971 \cdot \ 2.4495 \doteq 3.182 \doteq 3.182 \ \text{cm}^3$

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Pythagorean theorem is the base for the right triangle calculator.

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