one question

consider a square plate of a material whose coeficent of linear expan.is α
and that of superficial exp. is β.let the increase in temp be Δt
let original each side of square be a and the area be A

new length after increase in length = a(1+αΔt)

new area after increase in length = a²(1+αΔt)²

= a²(1+2αΔt)..(1)

new area = A(1+βΔt)...(2)

now A = a²

comparing (1) and (2) we get

β = 2α

its okay if you are desperate but theres no point in starting two threads for same question

if its your doubt
did you try to get the result by differentiation?

coefficients of linear , superficial and cubical expansion !