Transformations: Shape and Size

I've been thinking about this for some time and would like your views on the same. 2 questions..

1.

When we are comparing the size of two objects, does the shape have to be the same?

Suppose you have a circular plot and a rectangular plot of the same area, do they have the same size then?

Or can we even talk about the size of two objects of different shape? Suppose we have two objects of the same size, in that case, is it taken for granted that the 2 have the same shape as well?

2.

When we talk about size, what are we talking about?

For example if someone points to an object and asks what the size of it is. Is it the area, the volume or either of the 2 depending on the object?

Please put forth your views..

What I want specifically is, in transformations when we talk about preservation of size, is it the same as preservation of shape and size or is there a difference between the two? I understand that technically, it won't be possible to transform any shape into a different one using either of the transformations- reflection. translation or rotation and so talking about it might be futile but let's take a general look at size and shape.

Cheers!