When is it appropriate to use a circle graph to display data?

Usually called a pie graph or diagram. It is up to the presenter to chose the types of graphs that will best illustrate his ideas.

It would be inappropriate to show as a pie chart, a breakdown of activities that are not related. For example, two surveys ask different questions. The results should not be displayed on a single pie chart.

Generally, a means of visualizing the breakdown of a sum into logical subdivisions and given in percentages. For example, the breakdown of religious beliefs in the US in terms of percentages might make a good pie chart.

Pie charts should include the actual values. Also, there should be clear rules on what items are included in each group.

For example, for a particular company, one year 40% was spent on non-income generating activities and the next year, 15% was spent on these activities. One might get the impression that spending on these items was declining. However, one would also like to know in dollar terms, if spending on these activities was actually going up or down on non-income generating activities. One would also like to know the rules governing assignment of expenses to "non-income generating activities."

For more information, see:

http://en.wikipedia.org/wiki/Pie_chart