Hello, new to this forum stuff but I don't know where else to ask, but if I were to inscribe a regular Heptagon, Nonagon, and Undecagon in a circle with radius 1, how would I be able to find the points at which the vertices of the shapes touch the circle? I have one idea, to figure out at what angles the vertices are from the center and then use that and some trigonometric functions to find the exact values, but I don't know how to do any of that. Help?
Coordinate points of Polygons
- Thread starter Kryzix
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Dr.Peterson
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If this is for a course, please state the problem you were given, word for word. If not, please tell us the context of the problem -- why you are doing it, and what you will do with the answer. Either way, we need to be sure what you are given and what the goal is.
I suspect you may not mean "and", but "or" -- that is, that you want to inscribe any particular polygon, and to plot its vertices, not to intersect several of them at once.
I suspect you may not mean "and", but "or" -- that is, that you want to inscribe any particular polygon, and to plot its vertices, not to intersect several of them at once.
If possible, I would like one of the vertices to be at the point (0,1), on the upper side of the circle. But yes, the circle is at the origin.
This link might give you a better idea of what I'm doing.
Inscribing n-gons
www.desmos.com
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Cosine and sine give the horizontal and vertical coordinates, respectively, relative to the radius of a circle centered on the origin. As shown on the Wikipedia page for trigonometry:
If one of the vertices of an [MATH]n[/MATH]-sided polygon is at (1, 0), then the coordinates of the vertex [MATH]m = [0, n)[/MATH] are given as follows:
If one of the vertices of an [MATH]n[/MATH]-sided polygon is at (1, 0), then the coordinates of the vertex [MATH]m = [0, n)[/MATH] are given as follows:
[MATH]\theta = m\frac{360^{\circ}}{n}[/MATH]
[MATH]x = cos(\theta)[/MATH][MATH]y = sin(\theta)[/MATH]
[MATH]x = cos(\theta)[/MATH][MATH]y = sin(\theta)[/MATH]
This isn't for a course, I'm just doing this as a for-fun project on Desmos, where I have already made n-gons with sides 3, 4, 5, 6, 8, 10, and 12, but I really don't know what to do for 7, 9, and 11.
All I have really is a circle centered at the origin, with radius 1, and within it, inscribe n-gons 3-12. I would like to have one of the vertices be at (0,1) for the shapes that apply for continuity's sake (except for square, I made that one rotated to look more like a square and not a diamond (even though in this case they are the same).
This link might give you a better idea of what I am trying to do.
Inscribing n-gons
www.desmos.com
Dr.Peterson
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How did you get the coordinates for the other cases? You should be able to do exactly the same thing, unless you just looked them up somewhere!
Take what Mr. Bland said, but essentially swap x and y. For any n, each vertex is rotated 360/n degrees from the previous one; find the angle for a given vertex, then find the coordinates. Show us whatever you've got so we can commend or correct it.
Take what Mr. Bland said, but essentially swap x and y. For any n, each vertex is rotated 360/n degrees from the previous one; find the angle for a given vertex, then find the coordinates. Show us whatever you've got so we can commend or correct it.
pka
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Here is what I would do, but is requires knowledge of complex numbers.
For example the octagon is eight sides so look at the eight eighth roots of uinty.
The principal one is \(\displaystyle \phi=\exp\left(\frac{ik(2\pi)}{8}\right),~k=0\).
See the vertices HERE
.
At first I kinda just started guessing with the triangle and then used the distance formula to make sure all of the sides were of equal length once I thought I was close, then adjusted from there. I knew that the square had to have the same coordinates with different signs so I just found what that magic number was, and then from there I was able to get the hexagon, octagon, and dodecagon by graphing lines that are inverse to those of the sides of the shapes, found where those lines crossed the circle and marked those points, then connected them, and from there I was easily able to double the sides of a triangle to get a hexagon and again for a dodecagon. The same went for the square into an octagon. I found the pentagon online already constructed in Desmos so I just resized it to fit my radius and then did the same process to produce a decagon. But I already found a way to get the other shapes, I found a graph that has a circle and a line, and one can input the angle of the line so I set it to 90 (for the verticle orientation) and add 360/n to get the vertices of the shape I want, then I just connect the dots. Since I can do this very easily now, I don't see why I should stop at 12, might go for a higher number.
pka
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Absolutely correct that. Here are the vertices for a regular polygon with seventeen sides.
Please note that I used to same page that I first posted only changing 8 to 17 & 7 to 16.
Dr.Peterson
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If I were doing this, I'd want to find a way to enter n and have it calculate the locations of the vertices, rather than do it all by hand. Or, at least, make a spreadsheet to do all the calculations. Either way, you'd probably have to limit n to some range, since neither is (in itself) a programming language that could handle a variable number of variables.
@Kryzix, I've prepared a graph using @Dr.Peterson's suggestion, which includes a slider for the number of sides:
www.desmos.com
Desmos | Graphing Calculator
www.desmos.com
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