Consider a three dimensional body whose dimensions are l,b and h. Also the whole body is made up of a material whose co-efficient of linear expansion is given by α. 

The Coefficient of Volume Expansion and Coefficient of Linear Expansion Relationship

The coefficient of volume expansion (β) is a property of a substance that quantifies how its volume changes with temperature. It is defined as the fractional change in volume per degree Celsius change in temperature. The coefficient of linear expansion (α), on the other hand, describes how the length of a material changes with temperature. It is defined as the fractional change in length per degree Celsius change in temperature.

To show that the coefficient of volume expansion is three times the coefficient of linear expansion, we can use the relationship between volume and length.

1. Coefficient of Volume Expansion:
The coefficient of volume expansion (β) is defined as the change in volume per unit volume per degree Celsius change in temperature. Mathematically, it can be expressed as:
β = (ΔV/V) / ΔT

2. Coefficient of Linear Expansion:
The coefficient of linear expansion (α) is defined as the change in length per unit length per degree Celsius change in temperature. Mathematically, it can be expressed as:
α = (ΔL/L) / ΔT

3. Relationship between Volume and Length:
The volume (V) of a solid or liquid can be expressed in terms of its length (L) using the appropriate geometry. For a solid with cross-sectional area A, the volume is given by:
V = A * L

4. Calculating the Coefficient of Volume Expansion:
Substituting the expression for volume (V) in terms of length (L) into the definition of the coefficient of volume expansion (β), we get:
β = [(A * ΔL) / (A * L)] / ΔT
= (ΔL/L) / ΔT

5. Comparing Coefficients:
Comparing the expression for the coefficient of linear expansion (α) with the calculated expression for the coefficient of volume expansion (β), we can see that:
β = α

6. Conclusion:
From the comparison above, we can conclude that the coefficient of volume expansion (β) is equal to the coefficient of linear expansion (α). Therefore, the coefficient of volume expansion is not three times the coefficient of linear expansion.