polygons inscribed in circles

[DarkLink1994]

Hi
I am trying to calculate the left over area that the triangle makes with the pentagon which are both inscribed in a circle radius 1.
The interior angle of the triangle is 60 degrees and the interior angle of the pentagon is 108 degrees.

Please show how you work it out as I want to be able to apply this to n-sided inscribed polygons.

 

[pka]

I am trying to calculate the left over area that the triangle makes with the pentagon which are both inscribed in a circle radius 1.
The interior angle of the triangle is 60 degrees and the interior angle of the pentagon is 108 degrees.
Please show how you work it out as I want to be able to apply this to n-sided inscribed polygons.

If you do not show some work, we cannot help you.
We are not here to do your work for you.
We are more that happy to help you with what you have done.

 

[DarkLink1994]

Did you see the picture that I attached to the post?

 

[Dr.Peterson]

I am trying to calculate the left over area that the triangle makes with the pentagon which are both inscribed in a circle radius 1.
The interior angle of the triangle is 60 degrees and the interior angle of the pentagon is 108 degrees.

Please show how you work it out as I want to be able to apply this to n-sided inscribed polygons.

If I wanted to be able to apply the same technique to different n's (particularly if the ultimate goal was a program to do this), my first thought would be to use coordinates. There is an easy way to find the area of a triangle given its coordinates, and a straightforward (if not always easy) way to find the intersections of lines.

I'd also want to be aware from the start about possible complications in deciding which sides intersect, and which region I want the area of.

But what methods are you expecting to use? What have you tried? And what motivates the question, assuming it is not an assignment?

 

[DarkLink1994]

found the answer eventually i beleive ( see image)
K is number of sides of biggest polygon and n is number of sides of smallest polygon.

 

[Dr.Peterson]

Convince us it's correct. Or at least tell us what x and y mean.

 

[DarkLink1994]

Of course.

y is the 'left over' area after overlapping the polygons. In the picture I posted it is the area between the triangle makes that juts out of the pentagon.
The total area 'left over' in the diagram above between the pentagon and triangle will be 2y as there are two points of the triangle that jut out of the pentagon. It is the total 'left over' areas that I am interested in.

x is dependent on n and k. x has (n-1)/2 unique solutions. See eqn 1.

If you only look at prime sided polygons overlapped sharing the one identical 'origin point' then all vertices of the n sided polygon will be jutting out (except the origin point) See the 'primogram image'.

For overlapping prime sided polygons with a shared origin point you can add all the unique solutions of X up to find the total area 'left over' See eqn 2. You must multiply by 2 because there are two areas with the same unique solution.

I'll post about how I found the equation that caluclates the area in another post but for now thats all i've got.