Math Journeys, Volume 2
A Minor Intrusion, Part 3
In part 2 of this journey we had solved the problem of one regular pentagon containing exactly two sides of another in its interior and how their areas compare. The areas are in a ratio of about 2.618 to one. Here is what the situation looks like.
We extended the solutions to an analogous one involving triangles, squares and hexagons, and we summarized the results in a table. For details use the link below
We see that an increase in the number of sides results in an increasing disparity between the areas. In this final leg of the journey, we’ll see where this trend is going and apply a little bit of an advanced topic called limits.
Pushing On. At the end of part 2 it was suggested that a little bit of trigonometry and the formula for the interior angle of a regular polygon would help us solve for the general n- sided case.
We will skip the derivations of interior angle measure and the Law of Cosines The measure of each interior angle in a regular n-sided polygon is given by
(180n – 360) / n,
To see the formula in action we calculate the measure for a pentagon. Each angle is
(900 – 360) / 5, or 108 degrees.
We will apply the Law of Cosines to solve for c in the overlapping obtuse, isosceles triangle, as seen below in the case of pentagons.
We arbitrarily set the leg measures of the smaller pentagon to the unit length, and we label the obtuse angle at B with theta.The Law of Cosines resembles the Pythagorean Theorem, but it applies to any triangle.
The ratio c : 1 compares the sides of the pentagons, and c-squared : 1 compares the areas. The formula confirms last month’s result that the larger pentagon has approximately 2.618 times the area. For any number of sides the comparison of areas is c-squared, or
For example, dodecagons (12-sides) differ by a factor of 3.732.... To see what the long-term trend is we reform the angle expression
(180n – 360) / n = 180 – 360/n,
and that means that as n gets larger (many more sides), 360/n gets smaller, and the interior angle approaches a measure of 180, and that angle has a cosine of -1.
So for a very large n, the areas will approach a ratio of 4:1. That’s what we mean by a limit. We can increase the number of sides but the ratio is capped at 4.
A look at the dodecagon case will give us a sense of why we have a limit.
Even with only 12 sides, we see corners as flatter (approaching 180 degrees), and we see that the longer leg is about two of the shorter ones. A limiting side comparison of 2:1 means an area comparison of 4:1.
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