Volume of a prism

The volume of any prism is the area of the cross section multiplied by the length of the prism.

What is the volume of the triangular prism below? The cross section is an equilateral triangle.

First we need to work out the area of the cross section.

The area of a triangle is half the base \times the height.

The height of the triangle is shown by the dotted line and can be found by using Pythagoras’ Theorem

The height of the triangle can be calculated using

$h^2 = a^2 + b^2$ where $h = 2$ and $a = 1$

$2^2 = 1^2 + b^2$

$b^2 = 4 - 1 = 3$

$b = sqrt{3}$

Now we have all the measurements we need to calculate the area of the cross section of the prism and the volume of the prism

Let $A =$ the Area of the cross section

$A = \frac{1}{2} (base \times height)$

$= \frac{1}{2} (2cm \times sqrt3cm)$

$= sqrt3cm^2$

Let $V =$ the Volume of the prism

$V = A \times$ the length of the prism

$= sqrt3cm^2 \times 5cm$

$= 5sqrt3cm^3$

The volume rounded to two decimal places is $8.66cm^3$