# Volumetric dilation

As with superficial dilation, this is a case of linear dilation that happens in three dimensions, so it has a deduction analogous to the previous one.

Consider a cubic solids of sides that is heated a temperature , so that it increases in size, but as there is expansion in three dimensions the solid remains the same shape, having sides .

Initially the cube volume is given by:

After heating, it becomes:

When relating to the linear dilation equation:

For the same reasons as in the case of superficial dilation, we may neglect 3α²Δθ² and α³Δθ³ when compared to 3αΔθ. Thus the relation can be given by:

We can establish that the volumetric expansion coefficient or cubic It is given by:

Like this:

As for surface expansion, this equation can be used for any solid, determining its volume according to its geometry.

Being β = 2α and γ = 3α, we can establish the following relationships:

Example:

The circular steel cylinder in the drawing below is in a laboratory at a temperature of -100ºC. When it reaches room temperature (20 ° C), how much will it have dilated? Given that.

Knowing that the cylinder area is given by: