“A shape that is the set of all points on a plane equidistant from a given point (the centre).”
This effectively describes a circle as a shape having a constant radius. It is tempting to assume that a shape with constant diameter is therefore also a circle but this is not necessarily true. For example, a 50p coin has a constant diameter (so vending machines can identify them) but is clearly not a circle. If you don’t believe this, put a 50p on a piece of paper and draw two parallel lines that just touch it. Now rotate the coin. It will still fit between the lines. For a lot more on this see Kevin Houston’s blog
From a structural engineering perspective, is this anymore than a curiosity? Yes, for two reasons, one theoretical the other practical. From a theorectical perspective we often require structures to be circular and have specifications about just how accurately circular they should be. These are particularly important for very thin shell structures, such as silos or tanks, which can be very sensitive to small imperfections. Care must be taken to define definitions of “out-of-roundness” in terms of radius rather than diameter if certain types of imperfection are not to be missed (not all constant diameter shapes are as obviously non-circular as a 50p) . I have discussed this problem in full in this paper,
The related practical problem is one of measurement. Measuring diameters is easy – just put a tape measure right across. But measuring a radius is harder. Where is the centre? This is a problem that mechanical engineers have known about for sometime and they have well-developed means of measuring circularity based on radius for small objects. However, it is only recently the problem has been noted in structural engineering and care should be taken to measure correctly – a method is proposed here.