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May 18th, 2018, 06:19 AM   #1
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Number keeps coming up.

Hello,

I am new to the forum. I had something interesting happen while doing geometry and I wanted some input on it.

I was trying to find the area of a pentagon, and I was trying to find an equation for it. I found something very interesting in the process

I used a calculator online to find the area of a pentagon with a 3 foot radius and all equal sides. According to the calculator the area was 21.4 feet squared. I squared the radius and I divided 21.4 by the squared radius, which is 9 feet. The answer astounded me. I did the same thing with a different radius, this time 5 feet, and the same number came up.

The number was 2.37765.

No matter the radius and the area, if I square the radius and divide the area by the squared number, that number, 2.37765, shows up. I don’t know why.

My question is, why does this number keep popping up? Is it special somehow?

Jared
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May 18th, 2018, 09:11 AM   #2
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the area of a regular polygon in terms of radius and number of sides $n$ is given by

$A = \dfrac n 2 r^2 \sin\left(\dfrac{2\pi}{n}\right)$

and for a regular pentagon this is

$A = \dfrac 5 2 r^2 \sin\left(\dfrac{2\pi}{5}\right)$

dividing out the $r^2$ leaves

$\dfrac 5 2 \sin\left(\dfrac{2\pi}{5}\right) \approx 2.37764$

Last edited by romsek; May 18th, 2018 at 09:37 AM.
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May 18th, 2018, 10:35 AM   #3
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Formula for Area of a Regular Polygon

You really gave him an excellent answer using that formula involving the radius and sin(2pi/n) which I wasn't aware of. Also, in general as you probably know there is a simpler formula to determine the area of any regular polygon if you know its perimeter (p) or the length of each side and the length of an apothem (a) where an apothem is the perpendicular length from the center of the polygon to each side. An apothem is the same as the radius of a circle inscribed in the polygon whereas a radius of a polygon is the same as the radius of a circle circumscribed around the polygon. Anyway the apothem-perimeter formula is Area = (1/2)ap.
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May 18th, 2018, 10:55 AM   #4
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Hello,

I like looking at these equations. I really do. However, I have something very interesting that happened just now that I would like to share.

I basically made up a formula and tested it with an online calculator for a Pentagon. Let me explain. Let’s say there is a pentagon that has a 4 foot radius. According to the online calculator, The area of the pentagon is 38.04 Feet squared. My equation reaches the exact same answer, and it works for other Pentagon’s with a different radius. I have tested and proven it.

This is the formula.

Area=2.37765 Times the radius squared.

I did this with other radius lengths, like 5 feet, 20 feet, 6 feet, and using this equation I got the same answers as the Calculator. I have no idea why.

Do you think I am correct? Does this equation really work?

Jared
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May 18th, 2018, 11:32 AM   #5
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??? You asked this question in your first post and romsek immediately gave you a complete answer. Now you ask the same question again? What part of romsek's answer did you not understand?
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May 18th, 2018, 11:38 AM   #6
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Hello,

I asked, “why does this number keep popping up?” He gave me an equation for the area for a pentagon. My question was not answered.

Jared
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May 18th, 2018, 12:44 PM   #7
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Area of a regular pentagon = (5/2) * SIN(72) * r^2, where r is the radius

(5/2) * SIN(72) = 2.3776412....

Ya'll ok now?
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May 19th, 2018, 05:36 AM   #8
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What romsek told you first is a general formula for the area of a regular polygon with n sides and radius r.

$\text {Let } A(n,\ r) = \text { area of a regular polygon of } n \text { sides and radius } r.$

$\text {Let } f(n) = \dfrac{n}{2} * sin \left ( \dfrac{360^o}{n} \right ).$

$A(n,\ r) = f(n) * r^2.$

Therefore when you divide the area of ANY regular polygon with n sides by the square of its radius, you will always get the same result, namely whatever number f(n) is.

Romsek measured the angles in radians, but Denis used degrees, with which you may be more familiar. It makes no difference how you measure that angle provided you use the appropriate sine button on your calculator. In the case of a regular pentagon, where n = 5,

$f(5) = \dfrac{5}{2} * sin \left ( \dfrac{360^o}{5} \right) = \dfrac{5}{2} * sin(72^o) \approx 2.37764.$

Your mysterious number is simply part of the formula for the area of a regular pentagon

$A(5,\ r) = f(5) * r^2 \approx 2.37764r^2.$

EDIT: The numerical approximation of 2.37764 may appear arbitrary, but the exact statement

$\dfrac{5}{2} * sin(72^o)$

shows that it is related to the number of sides in a pentagon, namely 5, and the measure of the equal angles in a regular pentagon, namely 72 degrees. There is nothing arbitrary about it.
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Last edited by JeffM1; May 19th, 2018 at 05:45 AM.
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May 19th, 2018, 05:44 AM   #9
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(5/2) * SIN(72) = 2.3776412....

That could be nicknamed "pentagonPI", right Jeff

Another one for the crazies to extend into the billions!!
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May 19th, 2018, 06:00 AM   #10
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Quote:
Originally Posted by Denis View Post
(5/2) * SIN(72) = 2.3776412....

That could be nicknamed "pentagonPI", right Jeff

Another one for the crazies to extend into the billions!!
Actually that is an argument for using Romsek's radian measure. Now pentagonPI is even more obvious. We need to rewrite his formula slightly:

$A(5,\ r) = \dfrac{5}{2} * sin \left ( \dfrac{2}{5} * \pi \right ).$

With the reciprocals of 5/2 and 2/5 "cancelling out", we can see it really is just a manifestation of $\pi.$

I scare myself sometimes: that "cancelling out" sprang from my fingertips hours and hours before I had my first drink of the day.

We should (if we weren't so lazy) work it out for the volume of regular tetrahedrons and claim it as an ancient mystery of the Egyptians, who built the Pyramids as a clue to the mathematic nature of the universe.
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