The coefficient of areal expansion is a measure of the change in area of a solid per unit change in temperature. It is denoted by the symbol α. The coefficient of areal expansion is important in engineering and physics as it allows us to predict how a material will change in size with temperature variations.


The formula for the coefficient of areal expansion is given by:

α = (1/A) (dA/dT)

where A is the original area of the solid, dA is the change in area of the solid due to a change in temperature dT.


The coefficient of areal expansion can be derived using the relationship between linear expansion and area expansion. The linear expansion coefficient is denoted by the symbol β and is a measure of the change in length per unit change in temperature. The formula for linear expansion is given by:

ΔL = L0 β ΔT

where ΔL is the change in length, L0 is the original length, β is the linear expansion coefficient, and ΔT is the change in temperature.

The change in area of the solid due to a change in temperature can be expressed in terms of linear expansion as:

ΔA = L0 ΔL + (ΔL)^2

Substituting the value of ΔL from the linear expansion formula, we get:

ΔA = L0 β ΔT + (β ΔT)^2

Using the binomial expansion formula, we can simplify this expression to:

ΔA = L0 β ΔT (1 + β ΔT/L0)

Dividing both sides by the original area A, we get:

dA/A = β ΔT (1 + β ΔT/L0)

Taking the limit as ΔT approaches zero, we get:

dA/dT = β (1 + β ΔT/L0)

Dividing both sides by the original area A, we get:

α = (1/A) (dA/dT) = β (1 + β ΔT/L0)/A

Simplifying this expression, we get:

α = β (1 + α β)

Solving for α, we get:

α = β/(1 - α β)


In conclusion, the coefficient of areal expansion in solids can be derived from the linear expansion coefficient by using the relationship between linear expansion and area expansion. The coefficient of areal expansion is important in predicting how a material will change in size with temperature variations and is used in engineering and physics applications.