Regular hexagon, regular dodecagon HELP

[Ana.stasia]

I thought that since a regular hexagon consists out of six triangles that have all equal sides that the same would apply to the dodecagon. However while doing this question I realized it was not. Any explanation why a regular hexagon's triangles have all equal sides but regular dodecagon's don't? Why is it called a regular then?

 

[Deleted member 4993]

I thought that since a regular hexagon consists out of six triangles that have all equal sides that the same would apply to the dodecagon. However while doing this question I realized it was not. Any explanation why a regular hexagon's triangles have all equal sides but regular dodecagon's don't? Why is it called a regular then?
View attachment 22076View attachment 22077

Because the angles between the sides of a regular dodecagon, CANNOT be (60 * 2=) 120^o!!

 

[Ana.stasia]

Because the angles between the sides of a regular dodecagon, CANNOT be (60 * 2=) 120^o!!

So it needs to be 120 for the sides to be equal? Does that mean that a regular octagon can never have all equal sides either within its triangles?

 

[Deleted member 4993]

So it needs to be 120 for the sides to be equal? Does that mean that a regular octagon can never have all equal sides either within its triangles?

You CANNOT make regular octagon by centrally arranging "equilateral" triangles - as long as all the triangles are in one 2-D plane.

 

[Ana.stasia]

You CANNOT make regular octagon by centrally arranging "equilateral" triangles - as long as all the triangles are in one 2-D plane.

So there is not a regular octagon?

 

[pka]

I thought that since a regular hexagon consists out of six triangles that have all equal sides that the same would apply to the dodecagon. However while doing this question I realized it was not. Any explanation why a regular hexagon's triangles have all equal sides but regular dodecagon's don't? Why is it called a regular then?
View attachment 22076View attachment 22077

A regular inscribed polygon is a circle is one that has all of its edges of equal measure. The minimum numbers of sides is three.
It is a theorem that if there are \(n\) edges in such a regular figure then the angles between the sides measures \(\dfrac{(n-2)\cdot\pi}{n}\).
\(\left[\text{if one insists }\dfrac{(n-2)\cdot 180^o}{n}\right]\). Looking at your posted diagrams, with \(n\) vertices there are \(n\) isosceles triangles with their summit vertices the centre of the circle the measure of which is \(\dfrac{2\pi}{n}\); the sides forming the summit angles have measure the radius of the circumscribing circle; the measure of the base angles is one half of \(2\pi-\dfrac{2\pi}{n}\).

 

[Deleted member 4993]

So there is not a regular octagon?

Regular Octagon has all its 8 sides equal. It is NOT comprised of equilateral triangles.

Other regular polygons are made of ISOSCELES triangles (look at response #6 above)

 

[Dr.Peterson]

I thought that since a regular hexagon consists out of six triangles that have all equal sides that the same would apply to the dodecagon. However while doing this question I realized it was not. Any explanation why a regular hexagon's triangles have all equal sides but regular dodecagon's don't? Why is it called a regular then?
View attachment 22076View attachment 22077

A regular polygon is defined as one whose edges are all the same length (equilateral), and whose interior angles are all the same size (equiangular). It has nothing to do with the other lines you show, which are radii of the circle in which your polygons are drawn (the circumradius). Those segments are not considered part of the polygon itself.

 

[Ana.stasia]

A regular polygon is defined as one whose edges are all the same length (equilateral), and whose interior angles are all the same size (equiangular). It has nothing to do with the other lines you show, which are radii of the circle in which your polygons are drawn (the circumradius). Those segments are not considered part of the polygon itself.

Okay, I understand that. My question now is how do I calculate the angle that in a hexagon is 120 and in a dodecagon 150?

 

[Deleted member 4993]

Okay, I understand that. My question now is how do I calculate the angle that in a hexagon is 120 and in a dodecagon 150?

Look at response #6

 

[Dr.Peterson]

Okay, I understand that. My question now is how do I calculate the angle that in a hexagon is 120 and in a dodecagon 150?

If you don't want to memorize the formula shown in #6, you can obtain it from the isosceles triangles you drew.

For the dodecagon, there are 12 apex angles meeting in the middle, so each is 360/12 = 30 degrees. The interior angle of the dodecagon is the sum of two base angles of an isosceles triangle, namely 180 - 30 = 150 degrees.