20 points are equally spaced on the circumference of a circle. David draws all possible chords that connect pairs of these points. How many of these chords are longer than the radius, but shorter than its diameter? Let's first draw this out. Each of the 20 points is connected with the 19 other points on the circumference, forming 19 chords. One of these 19 chords is exactly equal to the diameter, leaving 18 chords to consider. For a chord to be longer than the radius, its central angle has to be larger than 60 degrees because an equilateral triangle with one of the vertices at the center of the circle means the chord is the length of the radius. Since the 20 points form a regular 20-gon, the central angle of the chord connecting two adjacent points is 18 degrees. Therefore, the angle needs to be at least 4 times that. Thus, three pairs of chords from a point to its nearest points along the circumference on either side are going to be shorter than the radius. This leaves us with 6 pairs of chords, or 12 chords per point. Since each chord is counted twice, once per endpoint, the total number of chords satisfying our condition is 20 times 12 and divided by 2 to cover the double counting, which gives us 120.

circle

chords

geometry

central angle

equally spaced points