# How this equation is attained of change of cross sectional area as thermal expansion occurs?

I've may been asking the very easy question.

The conductor of cylinder shape exists with the following parameters.

$$\alpha:=$$linear coefficient of thermal expansion of the conductor.

$$l_i:=$$length of the conductor as $$i$$ th state.

$$S_i:=$$cross sectional area of the conductor as $$i$$ th state.

$$t_i:=$$temperature as $$i$$ th state.

$$i\in \{1,2\}$$

Assumed that $$1$$st state is pre and $$2$$nd state is post.

$$l_2=l_1\{1+\alpha(t_2-t_1)\}$$

The problem is below equation.

$$S_2=S_1*\{1+\alpha(t_2-t_1)\}^{2}$$

I've googled to find out the formula of it but I've been unable to find it.

How this equation is attained?

Can anyone tell me some hint(s) or the website which describes it? so that I can deduce it in my own.

The cylinder has the same coefficient of expansion in all directions, so if the length increases by a factor of $$1+\alpha(t_2-t_1)$$, the area will increase by that factor squared.
To derive it yourself, let's say that the cylinder initially has radius $$R_1$$, so that $$S_1=\pi R_1^2$$. After expansion the radius will increase to $$R_2=R_1(1+\alpha(t_2-t_1))$$. Calculate $$S_2$$ and you will find it has the relation you give with $$S_1$$.