Part 2 of this post. I struggled for a day over this.
First, write down what you know...
1. Any regular polygon can be split up into equal isosceles triangles, one for each side. The angle in the middle is 360 divided by number of sides and the other two are just (180 deg minus the middle angle) divided by 2. The area of a triangle = height (known as apothem is this context) x base ÷ 2. Times that by the number of sides, and that's the total surface area.
Worked example with a square with side length 10:
Divide the square into four triangles, the angle in the middle is 90 deg and the other two angles are 45 deg. (45 deg is also half the inside angle of the corner of a square).
Each triangle has base length 10 and height/apothem 5, area = 5 × 10 ÷ 2 = 25.
25 × 4 side-triangles = 100.
2. According to the internet, "The area of any regular polygon is given by the formula: Area = (a x p)/2, where a is the length of the apothem (the line joining the half way point of a side and the centre) and p is the perimeter of the polygon", which is saying the same thing.
3. A two and a half sided regular polygon can divided into two and a half side-triangles with an angle in the middle of 144 deg (360/2.5). The corner angles are ((n-2)*180/n) = 36, so these triangles have half that = 18 deg. Check: 144 + 18 + 18 = 180, so far so good. Using cosines and tangents and a side length of 10, the height/apothem of each triangle is 1.625, so the area of each triangle is 1.625 x 10/2 = 8.125. It has two and a half sides, so the total area = 20.3125.
4. The surface area of a three-sided regular polygon (= an equilateral triangle) with side length 10 is 43.3, and the surface area of a regular two sided polygon (= a straight line) is zero, so 20.3125 looks about right to me.
"What's so difficult about that?" asks the crowd. Nothing in itself, it took me ten minutes, but...
5. A regular five pointed star (which appears to have the same shape as a two and a half sided regular polygon, see previous post) with side length 10 has surface area of 31.027 (handy calculator here). The area of the mini-pentagon in the middle is 9.59 (using this and this).
How do I reconcile 20.3125 with 31.027?
7. If we do the calculation in 3 but using five x side-triangles (because it appears to have five sides, even though we 'know' it doesn't), that means area 5 x 8.125 minus 9.59 (to subtract the overlaps) = 31.035, pretty close to 31.027 shown above (not sure why there's a small difference).
8. Or we start with 20.3125 and assume that the two and a half sided regular polygon has no pentagon in the middle (because it has no real corners), 20.3125 plus 9.59 = 29.9, which is even further from 31.027.
Hmm. Which explanation is more likely, 7 or 8? Or neither? Is it perhaps the case that the two and a half sided regular polygon is not 2-dimensional so can't be done using normal geometry, and if so, does it have more or fewer dimensions?
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2 comments:
I think the five-pointed star is a distraction and the surface area of a two and a half sided polygon is the area of the soap film that you would get if you made one out of wire and immersed it in soapy water. However this would be an approximation, because there is no way you could get the wires to cross each other in the middle, round the pentagon, without touching, which they obviously shouldn't.
B, I can easily imagine such a physical object. You just twist the pentagram (made out of coat hanger wire, for example) slightly so that the sides do not touch except at the outer corners.
Any attempt to get it to make a continuous soap bubble would fail - where one side passes through the plane formed by other corners. As the whole surface is a single surface like a kind of bizarre Moebius strip, it would then *pop*.
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