What is the difference between 'circular arc' and 'sector of circle' here...

In my maths formula booklet, for centre of mass, I am told:

Circular arc, radius r, angle at centre 2α: (r(sinα))/(α) from centre

Sector of circle, radius r, angle at centre 2α: (2r(sinα))/(3α) from centre

But what is the difference? In what situation/shape do I use circular arc formula, and what situation/shape do I use sector of a circle?

A good explanation is all I'm looking for.

Thanks very much.

Imagine the circle is a cake and you cut a usual slice which looks like a triangle but with two straight sides and a curved third side.

The sector is the whole slice.

The circular arc is the length of the curved edge