My problem of the week?

Question

A convex polygon can be defined as a polygon with all its interior angles less than 180 degrees. Also, all diagonals lie entirely inside a convex polygon. If a convex polygon has 324 diagonals, how many sides does this polygon have?

Answer 1

A diagonal joins any two vertices.

If we have n sides, we also have n vertices. Each vertex connects to n-1 other vertices. Two of those connections though are for the "sides", so each vertex has n-3 diagonals.

All told we have n(n-3) connections, but this is double counting all connections, so we have n(n-3)/2 diagonals.

n(n-3)/2 = 324
n^2-3n = 648
n^2 -3n-648 = 0
(n-27)(n+24) = 0

We have 27 sides to this polygon.

Answer 2

you need a formula for the number of diagonals. look at a square, 4 points. each of the 4 points can connect to 3 others, 3(4) = 12. but if you look carefully, that counts each connection twice. so 3(4)/2 = 6. 4 of those are sides, other 2 are diagonals. so in a convex n-gon, there will be n(n-1)/2 - n diagonals.

n(n-1)/2 - n = 324
(n² - n)/2 - n = 324
n² - n - 2n = 648
n² - 3n - 648 = 0
(n - 27)(n + 24) = 0
n = 27

Answer 3

25?