The Heptagon
The Heptagon (7 sided polygon) has been a shape of much mystery in geometry. It is impossible to construct a heptagon with compass and straightedge only. There are however many approximations, several of which I volition feature hither.
 | This first construction is the easiest. It is a very good approximation of the heptagon. Given circle O (yellow): - Observe A, a random betoken on the circle.
- Find the M, the midpoint of OA.
- Draw a perpendicular line through M. Information technology intersects circle O at B.
- Depict a circle at B (cyan) and then that it crosses K. Circle B intersects circumvolve O at two points of the heptagon. Yous tin can use them to find the rest.
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There is a method of approximating most any regular polygon with compass and straightedge. Information technology is especially useful with odd shapes like heptagons and nonagons. They are hard in that they involve the partition of lines. - Draw circle O. Draw AB, the diameter of circumvolve O.
- Divide AB into north number of parts (n being the number of sides , in this case seven!).
- Describe two circles, one at A and ane at B (cyan), each with radius AB. They intersect at C.
- Point D is the 2nd betoken from the left of the diameter.
- Describe a line from C to D, extending it all the style to the original circle, where it intersects at Eastward.
- A and E are two points of the heptagon. Utilise them to find the rest.
- You can utilize this approximate any n-gon!
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 | This next structure was formulated by myself. It's not equally accurate at the others, but information technology'due south relatively easy. - Draw circle O. Draw OA and OB, so that they are perpendicular to each other and A and B are on circle O.
- Describe BC, and so that it is perpendicular to AB (purple) and has half it's length.
- Draw a line through CA (cyan).
- Depict a circle at C then that it crosses B (red). Circumvolve C (cherry-red) crosses CA (cyan) at D.
- At point A, draw a circumvolve that crosses D. Circle A intersects circle O at two points of the heptagon.
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Here'south a very complicated construction, nevertheless very accurate. - Draw circle O. Depict ABCDE, a pentagon inscribed in circle O.
- Depict another circle at O, this time within the pentagon (purple). To exercise this, detect the midpoint of one of the sides of the pentagon and draw a circle crossing that point.
- Depict OA. OA intersects the small circle (imperial) at F.
- Draw a circle at F (cyan) that crosses A. It crosses OA at H. Draw a circle at A (cyan) that crosses H. Circle A (cyan) crosses OA at 1000.
- Describe a small circle at O (xanthous) that crosses H. Inscribe an equilateral triangle, HIJ within that circle (yellow).
- Draw a line through points I and J.
- Draw a circle at O that crosses point One thousand. This circle crosses IJ at points One thousand and L.
- K,Grand,L are three points of the heptagon. Employ them to detect the rest!
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See Nexusjournal for details |
 | This is a new construction. It involves the apply of a grid. - Notice point (2,iv). Draw a circle centered at the origin crossing that betoken.
- Draw a line at y = -1. This line crosses the circle at ii points of the heptagon.
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Shown here is the Neusis Construction for a Heptagon, which involves a marked straightedge. You tin find more than info at Mathworld. - Take a straightedge (such as a piece of paper) and marking 2 points on it, A and B.
- Construct segment CD which is equal to AB.
- Find the midpoint of CD and call it 1000. Draw a perpendicular bisector through indicate M.
- Draw segment CE and so that CE is perpendicular to CD and is equal in length.
- Draw a circle centered at D and crossing point Due east.
- Take your marked straightedge and place it so that A is touching the arc, B is touching the perpendicular bisector of CD, and the straightedge is touching bespeak C.
- So angle CBM or q is equal to p/14. Thus, 4q will give you 2p/7, which is what nosotros wanted!
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© April.7 2003 Robin Hu. All Rights Reserved. 
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