Central Angles and Intercepted Arcs - Problem 4

This is a problem that students commonly get incorrect because they use the wrong formula. I’m going to tell you that ABCDE is regular. Again you can’t judge anything in geometry just by how it looks, so regular means that all the sides are the same and all the angles are congruent.

I’m going to ask you to find the measure of arc BC. Now what most students do when they get it incorrect is they find the interior angle of whatever polygon I give them. But what you’re going to do, since you want to get the correct answer, is you’re going to find the center of that circle and you’re going to draw in some radii forming central angles.

So you’re going to say, well this is the arc that I’m concerned about and that arc will be congruent to that central angle and since we have a regular polygon, all of these angles are also congruent.

So to find the measure of this angle, we need to divide 360 by however many angles we have. If I know this angle then I can find the measure of the arc. So we have five angles here so 360 divided by 5, which I just want to make sure that I’m not going to mislead you here is 72. So I see that this angle is 72 degrees and the intercepted arc is congruent to its central angle. So measure of arc BC is 72 degrees.

The trick here was remembering that we’re looking for an arc and not one of these interior angles.