H  E  P  T  A  G  O  N

The most baffling shape in all Geometry.
By Robin Hu

Construction Number One

This is the Easiest Construction

Step 1: Draw an arbitrary circle centered at O. Draw OA, the radius of circle O.
Step 2: Draw a circle centered at A with a radius of length AO. Circle A intersects the original
circle at points B and C.
Step 3: Draw a line through BC.Lines OA and BC intersect at D.
Step 4: Draw a circle centered at B with a radius of BD. This circle intersects circle O at E.
Step 5: BE is a side of the Heptagon, use it to find the others.

Out of all the constructions, in my opinion, this first one is the most easiest and most accurate. Amazingly, this construction can only be found on My website. If you bisect one of the sides, you can create an approximation for a 14-gon.


Construction Number Two

Golden Ratio Heptagon

Step 1: Draw a circle A. Draw segments AB and AC so that AB is perpendicular to AC.
Draw a line through BC.
Step 2: Draw line BD so that BC is perpendicular to BD and BD is one half of BC.
Step 3: Draw a line through DC. Draw a circle at D with radius DB. Circle D intersects DC at E.
Step 4: Draw a circle at point C with radius CE. This circle intersects the original circle at two points of the heptagon.

This construction on the other hand, was actually invented by me. I found it by taking the Golden Mean of a side of square. Quite accurate for an amatuer construction.

Construction Number Three

Divide the Diameter into seven parts.

Step 1: Draw circle O with diameter AB drawn across it.
Step 2: Divide AB into 7 equal parts. Call the second point from the left P.
Step 3: Draw two circles, one centered at A and one centered at B.Both circles should have a radius of diameter AB. The two circles intersect at C.
Step 4: Draw a line through CP. Line CP intersects circle O at D. AD is a side of the heptagon.

This construction is actually based on the construction of an n-gon.

Construction Number Four

Construction Using A Pentagon & Triangle. The most accurate

Step 1: Draw circle O. Inscribe a pentagon ABCDE inside circle O.
Step 2: Draw a circle inside of the pentagon, tangent to all sides of the pentagon (Find the midpoint M of AB and draw a circle at O with radius OM).
Step 3: Draw a line through OA. OA intersects the tangent circle at F. Draw a circle at F with radius FA. Circle F intersects line OA at H.
Step 4: Draw a circle at A with a radius of AH or copy the length of AF twice over line OA. AG is twice the length of AF.

The most accurate and difficult construction

Step 5: Draw a circle at O with radius OH. Inscribe an equilateral triangle HIJ within this circle. Draw a line through IJ.
Step 6: Draw a circle at O with a radius of OG. This large circle intersects line IJ at points P and P2.
Step 7: Points G,P, and P2 are points of the heptagon.

See the NexusJournal website for more details.


Constructing a heptagon using a grid

I have included a new, and totally different construction for the heptagon. It involves the use of an equally spaced x-y grid.
It's very simple. Draw a circle, with the center (0,0) and intersecting the point (2,4) and (-2,4) on the grid. Make one of the points of the heptagon the point where the circle intersects the y-axis. Then, where the line y= -1 intersects the circle, draw two more points. There are now three points of the heptagon. You can now find the other 4 points easily.

Angle Errors
Construction Angle Measure Percent Error
Regular Heptagon 51.428571° 0.000%
Construction #1 51.317812° 0.215%
Construction #2 51.827292° 0.775%
Construction #3 51.518222° 0.174%
Construction #4 51.460483° 0.062%
Construction #5 51.4605° 0.062%

Email me if you have questions or comments.

More Pages
Square Triangle and Hexagon Pentagon Golden Ratio Nonagon
Pentadecagon Heptadecagon N-Gon Squaring a Circle Basic Constructions

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