How to Construct Regular Polygons Using a Circle

Constructing regular polygons accurately is very significant in geometry and is easy to do. If you have ever wondered about how to construct regular polygons from a circle, you’re reading the right article.

Method 1

Method 1 of 2:

Using A Protractor

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This will be the center line of your circle (dividing it into semi-circles).
Mark the center point.
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^{[1]XResearch source}
6
Calculate the angle between adjacent vertices, α. Since a circle has 360°, divide 360° by n, the number of vertices (or sides) to get α.^{[2]XResearch source}
- α=360°/n
- α is the measured angle between lines drawn from the center of the circle to adjacent vertices.
- For a dodecagon, n=12. A dodecagon has 12 sides and 12 vertices, so 360° divided by 12 is 30°, and α=30°.
Using the protractor, mark on the circumference of the circle all the multiples of angle α calculated above.^{[3]XResearch source}
8
Join the points marked on the circle with a line segment.^{[4]XResearch source} For a dodecagon there should be 12 marks and 12 sides, because it has 12 vertices. Don’t overlap the line segments.
- If your points are outside of the circle, then simply mark another point along the radial line from the center onto the circle for each point and then join them.
If they are, you can erase the circumscribed circle.

Method 2

Method 2 of 2:

Using A Compass, Ruler and Calculator

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Set your compass to the radius, r, and draw a circle.^{[5]XResearch source}
2
Calculate the length, ℓ, of each side of the regular polygon of n sides.
- ℓ=2*r*sin(180/n)^{[6]XResearch source}
- 180/n is in degrees, so make sure your calculator is set for degrees, not radians.
Be ultra-accurate and triple-check the measurement to ensure that it is as accurate as possible.
Don't change the radius of your compass.
5
Mark another arc or line on the circle. Continue the process until the arc or line touches the first point.
- Make sure your compass doesn't move!
6
Join the lines/arcs accurately using a ruler.
- Check to see that the sides are identical in length.
- If they are, then you're finished. Erase the construction guide lines.

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Question
How can I obtain angle 32.75 on a protractor?
Donagan
Top Answerer
With a normal protractor, all you can do is measure extremely carefully.
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If it is a pentagon, can I still use the compass method?
Donagan
Top Answerer
Yes.
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How do I construct an eleven-sided pologon
Donagan
Top Answerer
Use Method 1 above, except that the central angles used would not be 30°, they would be 360° divided by 11, or 32¾°.
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Tips

If you use a mechanical pencil, rotate the pencil slightly as you draw. This will produce a strong, consistent line. Otherwise, the lead wears and the line increases in boldness/width.
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For the final construction, do all of the marks in fine-line black pen, then paperclip a piece of tracing paper to your ink drawing and do the tracing carefully in pen or pencil.
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