SOLUTION: I want to make an octagon with all sides equal and all I have is a inside diameter. How can I figure this?

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Question 1190673: I want to make an octagon with all sides equal and all I have is a inside diameter. How can I figure this?
Found 3 solutions by Alan3354, greenestamps, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
I want to make an octagon with all sides equal and all I have is a inside diameter. How can I figure this?
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Make what, a drawing?
Or a solid?

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


A diagram....



The information you have is the diameter of the octagon, which is AE in the diagram. The objective (presumably) is to determine the length of the side of the octagon in terms of the length of the diagonal.

We can solve the problem the other way around -- finding the length of the diagonal in terms of the length of a side of the octagon.

For simplicity, we can let the side length of the octagon be 1. So AB=BC=CD=DE=1; and PQ is also 1.

Triangles AQB and CPD are 45-45-90 right triangles, so AQ=PD=sqrt(2)/2.

Then in triangle ADE the legs are lengths 1 and 1%2Bsqrt%282%29, and the hypotenuse is the given diameter.

Using the Pythagorean Theorem on that triangle...

d%5E2=1%5E2%2B%281%2Bsqrt%282%29%29%5E2

Use decimal approximations and a calculator gives us d=2.613 to 3 decimal places.

So, given the diameter d, the side length of the octagon is d/2.613 = 0.383d.

Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.

            From the context, you want to construct a  REGULAR  octagon.

            This polygon has a central angle of 45 degrees, which is exactly  HALF  of the right angle.

            The geometric construction procedure is as follows: 


(1)  If you have a diameter, you have a radius, too.


        (you find the radius by dividing the diameter in two equal part:

         - - - by bisecting the diameter.  There is a special standard geometric construction for it, using a compass and a straightedge).



(2)  Having the radius, you draw a circle of this radius.



(3)  Then you draw an arbitrary diameter through the center, using the straightedge.



(4)  Having the diameter drawn, you construct a perpendicular to it through the center, using the compass and the straightedge.



(5)  Having this right angle, you divide it in two halves using the compass and the straightedge.


     In other words, you bisect this right angle.  There is a special standard procedure for it,

     using the compass and the straightedge.



(6)  Doing in the same way, you divide all right angles in two halves, drawing their bisectors.


     Doing this way, you obtain 8 intersection points on the circle, that are vertices of your octagon.


     The final step is to connect the consecutive vertices to get your figure.


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Below are the links to my lessons in this site, describing standard geometric constructions

    - HOW TO bisect a segment using a compass and a ruler

    - HOW TO bisect an angle using a compass and a ruler

    - HOW TO construct a triangle using a compass and a ruler