Suppose a sheet of a solid with surface area A₁ at temperature T₁ is heated to temperature T₂ such that ΔT = T₂ – T₁ is very small. Let A₂ be the surface area of the sheet at temperature T₂ . Experimentally, it is found that the increase in the surface area of the sheet (areal expansion), A₂ – A₁ , is proportional to A₁ and ΔT. Therefore,

(A₂ – A₁ )α A₁  ΔT

∴ A₂ – A₁ = σ Al₁ΔT, where a is the constant of proportionality, called the coefficient of areal expansion of the solid.

σ = \(\cfrac{A_2-A_1}{A_1ΔT}\), It is expressed in per °C.

We have A₂ = A₁ + σA₁ ΔT = A₁ (1 + σΔT).

σ is the increase in the area of a solid per unit original area per unit rise in its temperature. [Note: Consider a thin square metal plate of length l. Area of one face of the plate = A = l₂ . Suppose the plate is heated so that the rise in its temperature is ΔT (assumed to be very small). Then in the usual notation, Δl = l λΔT and ΔA = AσΔT = l² σΔT. Also, ΔA = (l + Δl)² – l² = l² + 2l.Δl + Δl² – l2 = 2l.Δl + Δl² . As Δl << 2l.Δl^2, we can write

ΔA = 2l.Δl(approximately)

∴ ΔA = 2l(l λΔT) = 2l² λΔT but ΔA = l² σΔT

∴ σ = 2.λ]