Thank you to Lucy from Caversham Primary School, Ruth from Swanborne House, Keshvi from Akthorpe Primary and Katy from Belilios Public School who all sent in clear solutions. Lucy explained her reasoning very clearly:

The triangle can go in three ways:
black dot at the top point
black dot at the left point
black dot at the right point

The square can go in four ways:
black dot in top left corner
black dot in top right corner
black dot in bottom right corner
black dot in bottom left corner

However, the circle has no corners so it can go in a countless number of times.

Katy adds:

The regular triangle has rotational symmetry of order 3. So, it can be rotated 3 times.
In the same idea, square has rotational symmetry of order 4. So it can be rotated 4 times.
At last, for a circle, there's infinity order of rotational symmetry. So, there are an infinite number of ways for a circle to rotate.