Thermal expansion
Most substances expand when heated, and the amount of expansion depends on the material and on the change in temperature. Expansion can be described as a change in length (linear expansion) or a change in volume (volume expansion).
Linear expansion
When a solid object is heated from temperature $T_1$ to $T_2$, its length changes. The change in length is proportional to the original length and to the change in temperature:
$\Delta \ell=\alpha_L\, \ell_0\, \Delta T$
so the new length is
$\ell=\ell_0\left(1+\alpha_L\, \Delta T\right)$
where $\ell_0$ is the original length, $\Delta T=T_2-T_1$, and $\alpha_L$ is the coefficient of linear expansion (a property of the material).
Volume expansion
When an object is heated, its volume also changes. The change in volume is proportional to the original volume and to the change in temperature:
$\Delta V=\alpha_V\, V_0\, \Delta T$
so the new volume is
$V=V_0\left(1+\alpha_V\, \Delta T\right)$
where $\alpha_V$ is the coefficient of volume expansion. For a solid, $\alpha_V\approx 3\alpha_L$.
Example 1: The steel arch of the New River Gorge Bridge in West Virginia spans about $518\text{ m}$. On a cold winter day the steel is at $-12\,^{\circ}\text{C}$, and on a hot summer day it reaches $32\,^{\circ}\text{C}$. How much longer is the span on the summer day? (For steel, $\alpha_L=12\times10^{-6}\,/^{\circ}\text{C}$.)
Solution: First find the change in temperature:
$\Delta T=32\,^{\circ}\text{C}-(-12\,^{\circ}\text{C})=44\,^{\circ}\text{C}$
Now use $\Delta \ell=\alpha_L\, \ell_0\, \Delta T$ with $\ell_0=518\text{ m}$:
$\Delta \ell=(12\times10^{-6})(518)(44)=0.27\text{ m}$
So the steel span is about $0.27\text{ m}$ ($27\text{ cm}$) longer on the hot summer day than on the cold winter day.
Example 2: If you heat a thin circular ring, does the ring's hole get larger or smaller?
Solution: The hole gets larger. When an object is heated, every linear dimension increases by the same factor, as if the object were enlarged in a photograph. The hole expands just as the material around it does.