An object of dimensions L_o, W_o and H_o has a volume:

V_o = L_o W_o H_o

When the temperature changes the volume changes to:

V = L_o W_o H_o [1 + αΔT]³

V = V_o [1 + 3αΔT + 3(αΔT)² + (αΔT)³]

αΔT is much less than 1, so (αΔT)² and (αΔT)³ are negligible. So:

V = V_o [1 + 3αΔT]

V = V_o [1 + βΔT]

where β, the coefficient of volume expansion, is three times larger than α the coefficient of linear expansion.