Octagons for Geometry Dummies

Octagon geometry

One sign of a freelancer who's bullshitting readers, especially at a place where he or she is supposedly "answering questions," is an obvious inability to explain the reasoning behind an approach within the "answer." Our research team members see that quite often at the niche sites where old eHow.com articles go to die (compliments of Leaf Group), and today's no exception. Leafer Chance E. Gartneer (wonder what his real name is) screwed the pooch in "How to Find the Length of the Sides of an Octagon Based on the Diameter," which nowadays can be found over at Sciencing.com.

Our staff math guy thought this was an interesting question, since there are two answers because any regular octagon has two different diameters: one for the circle that touches all the points (circumscribed circle) and a slightly smaller one for the circle tangential to the midpoints of all the sides (inscribed circle). Either way, it just takes a little trigonometry to figure it out. Chance sort of gets to that...

If you have the diameter of the inscribed circle, Chance says to Calculate the value of π/8 Calculate the tangent of π/8 radians Multiply the diameter by the tangent to get the length of a side... ...and, indeed, that works. He also explains using the diameter of the circumscribed circle... and gets a right answer (in step 3, you multiply by the sine of π/8 instead of the tangent). Our problem is not that "Gartneer" got the wrong answer, our problem is that he seemingly plucked some numbers and trigonometric functions out of thin air.

Chance's "tip" for students wanting to know why this even works is to use an "online regular polygon calculator" to check their work. That's not helpful, dummy: what would have been helpful is to explain why this works instead of packing all the calculations inside some black box. This works because the a) inradius (AKA apothem) is one leg of a right triangle and the other leg is half of a side, b) the circumradius is the hypotenuse of the same right triangle, and c) there are 2π radians in a circle, which means that π/8 radians is the angle between the inradius and the circumradius in a regular octagon.

We're almost certain that Chance had no earthly idea why his "solution" works (and neither did his vaunted content editor). That's why today's Gartneer is getting his fifth Dumbass of the Day award today, neatly packed away in a sealed black box.

copyright © 2017-2022 scmrak

MM - TRIGONOMETRY