Historically, circles are important because they were one of the first geometric shapes that early mathematicians had the ability to study analytically; i.e. graphically and algebraically. Analytic geometry has since expanded greatly, and now provides some of the mathematical foundation for such fields as topology and general relativity.
Additionally, the symmetric nature of circles have influenced philosophers and certain religions throughout the ages.
Let's also not forget that, without circles, we'd be driving on square wheels!
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A degree is one measure of angles. There are 360 degrees in a circle and that gives 360 its importance in relation to a circle.
No sharp corners to poke yourself on.
The diameter of a circle determines all its characteristics.
It is important to know formulas of pi for circles like areas of a circle and its circumference
2*1673 = 3346 [don't forget that units are important].