Ugh! I don't much care for that question because it depends on the units used to measure the angles. You will get a much different answer in radians than in degrees. So it's a very unnatural thing to ask.
How to solve it: A diagonal corresponds to a pair of nonadjacent vertices. There are "n choose two" or n(n-1)/2 pairs of vertices and n of those are adjacent. Thus n(n-3)/2 diagonals.

In radians, the sum of the angles is π × (n-2). So you need to solve the inequality

(n-2)π < n(n-3)/2

which you do by solving the corresponding equation, which can be rewritten as

2 × π × n - 4 × π = n² -3n

or in standard form

n² - (3+2 × π)n + 4 × π = 0

Plugging that into the quadratic formula gives about 7.637, so a heptagon should have fewer diagonals (14) than its total internal radian measure (5 π) while an octagon should have more (20 vs 6 π). And lo, it was so.

Now, if you want your answer based on degree measure [sum of angles is 180(n-2)], you know how to solve it. The answer will of course be much larger.

Good Hunting!
RD