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.
Contents
Steps
Using A Protractor
- Draw a straight line using the protractor. This will be the center line of your circle (dividing it into hemispheres).
- Align the protractor with both 0° and 180°on the center line, then mark the center point.
- Trace the semicircle along the protractor from 0 ° to 180°.
- Put the protractor on the other side of the center line, again with the both the 0°and 180° protractor markings on the center line.
- Complete the circle by tracing along the protractor.
- 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.
- 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 comes out to be 30° and α=30°.
- Mark a point for each of the successive angles. Using the protractor, mark all the multiples of the angle α calculated above.
- Join the points marked on the circle with a line segment. 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.
- Check to see that the sides are the same length. If they are, you can rub out the circumscribed circle.
- Finished.
Using A Compass, Ruler and Calculator
- Draw a circle of the desired radius, r. Set your compass to the radius, r, and draw a circle.
- Calculate the length, ℓ, of each side of the regular polygon of n sides.
- ℓ=2*r*sin(180/n)
- 180/n is in degrees, so make sure your calculator is set for degrees, not radians.
- Set your compass to this length, ℓ. Be ultra-accurate and triple check the measurement to ensure that it is as accurate as possible.
- Start from any point on the circle and mark an arc or line. Don't change the radius of your compass.
- 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!
- 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. Rub out the construction guide lines.
Tips
- For the ultimate 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.
- 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.
Things You’ll Need
Method One
- Protractor
- Calculator (optional, but maybe necessary depending on the number of sides)
- Paper
- Tracing paper (optional)
- Pencil
- Pen - black, fine point (optional)
- Eraser
Method Two
- Compass
- Ruler
- Calculator
- Paper
- Tracing paper (optional)
- Pencil
- Pen - black, fine point (optional)
- Eraser
