Perimeter of Regular Polygon

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nasi112

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Find the perimeter of a regular polygon of 180 sides inscribed in a circle of diameter 1.00000 m. Compare this number with the circumference of the circle.

I try to solve this question geometrically. I can't draw 180 sides, so I drew a few.

I found this.

circledpolygons.png
I need software name that can draw these shapes. What patterns I need to see here?

I'm not in hurry to solve it and I will read all comments. Thanks in advance.
 
Beer drenched reaction follows.
nasi112 said:
Find the perimeter of a regular polygon of 180 sides inscribed in a circle of diameter 1.00000 m. Compare this number with the circumference of the circle.

I try to solve this question geometrically. I can't draw 180 sides, so I drew a few.

I found this.

View attachment 39882
I need software name that can draw these shapes. What patterns I need to see here?

I'm not in hurry to solve it and I will read all comments. Thanks in advance.
Click to expand...

The nasi112 I remember from 4 or 5 years ago would have made short work of that little puzzle.
Are there two or more users of the nasi112 username?

Screenshot_20251211-002124_Drive.jpg
 
nasi112 said:
Find the perimeter of a regular polygon of 180 sides inscribed in a circle of diameter 1.00000 m. Compare this number with the circumference of the circle.
1765394832351.png
I try to solve this question geometrically. I can't draw 180 sides, so I drew a few.

I found this.

I need software name that can draw these shapes. What patterns I need to see here?

I'm not in hurry to solve it and I will read all comments. Thanks in advance.
Click to expand...
The Geogebra App can draw these shapes for you. But, as @jonah2.0 has already demonstrated above, there is no need to do so; the answers required can be arrived at algebraically.

If you insist on a need to draw the figures I have created a Geogebra construction that will allow you to do so.

Clicking on this link will open Geogebra with a triangle inscribed in a circle and a slider set to: n=3 (qv).

As you move the slider to increase n (the number of sides in the polygon) you will get a pentagon (for n=5), a hexagon (n=6) and so on.

You can export any of the images created to file (or your clipboard) by clicking on the hamburger (top right) and choosing 'Export' from the drop-down menu.

I have allowed the slider to go up to 200 but once you create an icosagon (20 sides) or above, the polygon becomes virtually indistinguishable from the circle! 🤷‍♂️
1765396636205.png




Here's what happens when n=180......




Hope that helps. 😊
 
Thanks guys.

jonah2.0 said:
Beer drenched reaction follows.


The nasi112 I remember from 4 or 5 years ago would have made short work of that little puzzle.
Are there two or more users of the nasi112 username?
Click to expand...
That's me:)

I left study more than 3 years in a row. Recently got back and covered infinite series. Went through probability as well.

The Highlander said:
If you insist on a need to draw the figures I have created a Geogebra construction that will allow you to do so.
Click to expand...
Thanks. I will check it out.
 
Beer soaked ramblings follow.
nasi112 said:
Find the perimeter of a regular polygon of 180 sides inscribed in a circle of diameter 1.00000 m. Compare this number with the circumference of the circle.

I try to solve this question geometrically. I can't draw 180 sides, so I drew a few.

I found this.

View attachment 39882
I need software name that can draw these shapes. What patterns I need to see here?

I'm not in hurry to solve it and I will read all comments. Thanks in advance.
Click to expand...

One of the them amazing definitions: The area of a circle is defined as the limit of the areas of
inscribed regular polygons as the number of sides increases without bound.

For a regular polygon with 4 sides

Screenshot_20251211-140512_Chrome.jpg,
 
There are quite a few fascinating formulas to approximate [imath] \pi. [/imath] The problem is old, very old: 1800 BC (Babylon).
I like Archimedes's result [imath] 3+\dfrac{10}{70}> \pi > 3+\dfrac{10}{71} [/imath] (250 BC) who used polygons. The most famous formula is probably the Wallis product [math] \pi=2\cdot \dfrac{2}{1}\cdot\dfrac{2}{3}\cdot\dfrac{4}{3}\cdot\dfrac{4}{5}\cdot\dfrac{6}{5}\cdot\dfrac{6}{7}\ldots [/math]However, Ramanujan wouldn't have been Ramanujan if he hadn't found a computationally really efficient approximation
[math] \dfrac{1}{\pi}= \dfrac{\sqrt{8}}{9801}\cdot\displaystyle{\sum_{k=0}^\infty \dfrac{(4k)!(1103+26390k)}{(k!)^4\cdot 396^{4k}}}.[/math]
 
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