Why can regular polygons be inscribed into a circumference?

Dean Summers 2022-07-20 Answered
Why can regular polygons be inscribed into a circumference?
Why can regular polygons be inscribed into a circumference? I have asked this to myself a lot of times. I have also wondered why all triangles can be inscribed into a circumference but I think that it is not so difficult to see why.
I also have another question. Suppose that you have a circumference of radius R and you draw a chord of length L in some direction. And then draw another circumference with the same radius and draw another chord with same length L but in another direction, is it possible to rotate the second figure or apply some transformation to it, so that it looks like the first figure? It might be a silly question but I'd appreciate any answer or help.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Brendon Bentley
Answered 2022-07-21 Author has 11 answers
Step 1
Take two consecutive sides of your regular polygon, as AB and BC in diagram below, and draw their perpendicular bisectors, intersecting at point O. The properties of the perpendicular bisector imply A O B O C O. It follows that triangles OAB and OBC are isosceles and congruent by SSS, and O B A O B C 1 2 A B C 1 2 B C D O C B O C D ..
Step 2
Consider now side CD, consecutive to BC. By SAS triangles BCO and BDO are congruent, hence D O C O and O C D O D C O D E. You can go on like that, to show that all vertices have the same distance from O.

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-08-18
Covering the plane with convex polygons?
I have got the following task here:
Prove, that you can't cover the "Plane" with convex polygons, which have more than 6 vertices!
The answer is pretty obvious for n = 3 vertices, because 6 60 = 360 .
For n = 4 it works too, because 4 90 = 360 .
I think that n = 6 is good too, but how do I prove, that other than that, it isn't possible to do that?
asked 2022-08-11
Constructible polygons
I know certain polygons can be constructed while others cannot. Here is Gauss' Theorem on Constructions:
cos ( 2 π / n ) is constructible iff n = 2 r p 1 p 2 · · · p k , where each p i is a Fermat prime.
Can this is be used to determine the constructibility of a regular p 2 polygon? If so how and what would be the cos ( 2 π / n ) here?
asked 2022-08-09
“Area metric” and “Hausdorff metric” are not equivalent on all closed polygons, but equivalent on convex closed polygons
Suppose X is the set of all closed polygons, d Δ is the “area metric” defined by the area sum of the symmetric difference of two closed polygons, and dH is the Hausdorff metric on X. How should I prove that d Δ and d H do not generate the same topology on X? Also why do they generate the same topology on the subset of convex polygons? I tried to visualize how a typical open ball in both metrics looks but this seems rather impossible.
asked 2022-08-07
The area of a triangular sail for a boat is 176 square feel. If the base of the sail is 16 feet long, find its height.
asked 2022-09-24
If I have a circle that has ten dots on it and I have to make as many polygons using those dots, the answer should be 10. Connecting each dot starting with triangle, to square, all the way to the max of a 10-sided polygon should give me only 10 polygons that I can make. Am I missing something here?
asked 2022-09-29
Relationship between area of similar polygons and their corresponding lengths
It is known that ratio of the areas of two similar polygons is equal to the square of the ratio of the corresponding sides. Could one advise me how to prove this assertion? Do we divide the polygon into triangles?
asked 2022-09-16
Does the intersection of two concave polygons without holes result in a set of several polygons without holes?
I did not find a proof anywhere, so here is my assumption :
If I have 2 concave and simple polygons without holes, then the intersection yields a set of different polygons without any holes.
If it can result to a polygon with a hole inside, could you provide an example?
Edit: the polygons are simple. edges of one polygone don't intersect, except consecutive edges, which intersect in their common vertex. The polygons in input do not have holes. I consider the intersection of the interiors of the polygons. The resulting set is all the points that are included both in interior of polygon P1 and the interior of the polygon P2.

New questions