Over small temperature ranges, the linear nature of thermal expansion leads to expansion relationships for length, area, and volume in terms of the linear expansion coefficient .
The relationship governing the linear expansion of a long thin rod can be reasoned out as follows:
Over small temperature ranges, the fractional thermal expansion of uniform linear objects is proportional the the temperature change. This fact can be used to construct thermometers based on the expansion of a thin tube of mercury or alcohol. Several equivalent forms of the relationship find use. The calculation is set up in the form