



 
Hi Tom, There is no reason that the central angle needs to measure less than 180 degrees. I think that one of the examples on Math Central that you might have found confusing is my response to Fleur who was constructing a lampshade. I have copied the images from that page so that I can easily refer to them. The green lampshade is on the left and the template to form it is on the right. I used the green colour to indicate which part of the image on the right is to be used to form the lampshade. When I made the calculations I was focused on the pink piece to be removed so the central angle I calculated was the central angle of the pink section which turned out to measure 142 degrees. I could just as easily have calculated the central angle of the green segment on the right which, as you observed, would be 360  142 = 218 degrees. Let me do that calculation. From my response to Fleur we know that $PR = PD = 33$ cm and $BD = 20$ cm. Thus the circumference of the base of the lampshade measures $2 \pi \times 20$ cm. On the template on the right this is the length of the arc measured counterclockwise from Q to R. Since $PR = 33$ cm the circumference of the outside circle on the right measures $2 \pi \times 3$ cm. Hence the central angle of the green sector on the right measures \[\frac{2 \pi \times 20}{2 \pi \times 33} \times 360 = 218 \mbox{ degrees.}\] I hope this helps,
 


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