# Geometry with polygons formed within drawn stars

• Omega234
In summary, the conversation discusses the construction of a 16-gon using a specific method and the existence of a smaller 16-gon within it. The individual is unsure if the interior 16-gon is regular and is seeking help in proving it. The suggestion is made to consider angles and prove that there are 16 of them spaced at equal central angles, which would indicate that the interior 16-gon is also regular.
Omega234

## Homework Statement

I'm working with the polygons on the interior of stars that have been drawn in a specific manner. For an example, I'm currently using a 16-gram. To construct the same one I have, you array sixteen vertices equidistant from one another around a central point (as in the construction of a regular hexadecagram). Starting at one point, connect it directly to the point that is five spaces from it clockwise (so there will be four points in between that have nothing through them). Continue with this pattern of fives until you eventually get back to the beginning.

Inside this star, another hexadecagram should exist. My question is whether this polygon is regular, and if so, how do you prove it? My intuition tells me that it ought to be nothing more than a smaller version of the hexadecagram that would be created by directly connecting each of the exterior points in a circular fashion, but I've learned well enough by now that 1) intuition is not always right, and 2) intuition is very difficult to cite in a paper.

## Homework Equations

Honestly, I'm not sure of any relevant equations here. That's sort of why I'm here, haha.

## The Attempt at a Solution

Well... I drew it. And it looks mildly regular-ish. That hardly constitutes a proof in itself, though. I'm pretty decent at algebra and calculus, but I've never been the best problem-solver when it comes to geometry, haha.

Mostly if someone could help me figure out an actual proof for whether or not the interior polygon is regular, I would be most appreciative. Any ideas? Thanks in advance!

Think about angles. Clearly each new line you are drawing has the same minimal radius to the center point of the original 16-gon. If you can prove that there are 16 of them spaced at equal central angles (think about how multiples of 5 divide into 16) wouldn't that prove the interior 16-gon is also regular?

## 1. What are the properties of polygons formed within drawn stars?

Polygons formed within drawn stars have the same properties as regular polygons. They have straight sides, vertices, and angles. The number of sides and angles will depend on the number of points in the star.

## 2. How do you calculate the area of a polygon formed within a drawn star?

The area of a polygon formed within a drawn star can be calculated by dividing the star into smaller triangles and using the formula A = 1/2 * base * height for each triangle. Then, the areas of all the triangles can be added together to get the total area of the polygon.

## 3. How do you determine the perimeter of a polygon formed within a drawn star?

The perimeter of a polygon formed within a drawn star can be found by adding the lengths of all the sides of the polygon together. If the star has n points, the perimeter can also be calculated by multiplying n by the length of one side.

## 4. Can a polygon formed within a drawn star be regular?

Yes, a polygon formed within a drawn star can be regular. A regular polygon is one in which all sides and angles are equal. Depending on the number of points in the star, it is possible for the polygon to be regular.

## 5. Are there any real-world applications of studying polygons formed within drawn stars?

Yes, there are many real-world applications of studying polygons formed within drawn stars. For example, architects and engineers use geometric shapes to design buildings and structures. The properties of polygons formed within drawn stars can also be applied in fields such as computer graphics, art, and even video game design.

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