Question number 11. Radu has identical plastic pieces in the shape of a regular pentagon. He glues them edge to edge to complete a circle, as shown in the picture. How many pieces are there in this circle? I have enlarged the diagram, and the first thing we should do is locate the center of that circle, which can be done by extending the edges of each pentagon until they meet at a point. And now we can see that the number of pentagons necessary to complete the circle is related to the measure of this angle over here, which is the same as the angle on the opposite side, and I'll call that alpha. So we have the number of pentagons equal to 360 degrees divided by the measure of alpha. And furthermore, we can say that alpha belongs to a isosceles triangle, the one right here, where the other angle, let's call that one beta. And so there are two of them like that. And we can say that alpha would be 180 degrees minus twice the measure of beta. So let's compute beta, and then we'll be able to finish the problem. Now, knowing something about regular pentagons, we know that each is made of five isosceles triangles, like this, where the central angle, that angle there, is 72 degrees, because that's 360 divided by 5. And so that tells us that each of the remaining angles measures 54 degrees. That allows us to compute the measure of the interior angle, which would be 108 degrees. And with that, we see that beta is its supplement. So beta is 180 degrees minus 108, and that gives us 72 degrees. From here, we compute alpha, which comes out to 36 degrees. And so finally, 360 degrees divided by 36 degrees is 10, and that's the number of pentagons needed to complete the circle.

regular pentagon

central angle

isosceles triangle

circle formation

geometry