Problem of the Week #5: Monday Feb. 6th, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems. Please refrain from posting full solutions.
In a regular polygon, an apothem is the radius of the incircle, or a line segment joining the center and the midpoint of a side, like this:
If we’re given the side length (s) of a polygon, it’s relatively straightforward to calculate the length of the apothem (a) using angles:
where n is the number of sides, which would give us:
It’s nice and clean apart from the fact that it involves a transcendental function where we need to rely on the calculator for an answer, which was fine with me until I saw this video.
Notice how the person constructing polygons from a given side length in this video took the apothems of a square and a regular hexagon and just… averaged them (or found the midpoint) to get the apothem of a regular pentagon. Not only that, the linear difference is then added on to find the apothems of a regular heptagon, octagon, and nonagon.
a). Is this theoretically accurate? Or is this an approximation?
b). If this is theoretically accurate, prove that for a given side length s,
i. the apothem of a regular pentagon is the average of the apothems of a square and a regular hexagon.
ii. the sequence of apothem lengths of regular polygons is an arithmetic sequence
c). If this is an approximation,
i. find the error margins for cases where 5≤n≤9.
ii. find the range of n where this approximation works reasonably well, for the purpose of drawing regular polygons by hand for example.
d). Share your problem inspired by this scenario.