

Regular hexagon, given one side
Geometry construction using a compass and straightedge
How to construct a
regular hexagon
given one side. The construction starts by finding the center of the hexagon, then drawing its
circumcircle,
which is the circle that passes through each
vertex.
The compass then steps around the circle marking off each side.
Printable stepbystep instructions
The above animation is available as a
printable stepbystep instruction sheet, which can be used for making handouts
or when a computer is not available.
Explanation of method
This construction is very similar to Constructing a hexagon inscribed in a circle,
except we are not given the circle, but one of the sides instead.
Steps 13 are there to draw this circle, and from then on the constructions are the same.
The center of the circle is found using the fact that the radius of a regular hexagon (distance from the center to a vertex)
is equal to the length of each side.
See Definition of a Hexagon.
Proof
The image below is the final drawing from the above animation.

Argument 
Reason 
1 
ABCDEF is a hexagon 
It is a polygon with six sides.
See Definition of a Hexagon. 
2 
AB, BC, CD, DE, EF, FA are all congruent. 
Drawn with the same compass width AF. 
3 
A, B, C, D, E, F all lie on the circle O 
By construction 
4 
ABCDEF is a regular hexagon 
From (1), (2). All its vertices lie on a circle, and all sides are congruent.
This defines a regular hexagon. See
Regular polygon definition and properties 
 Q.E.D
Try it yourself
Click here for a printable worksheet containing two problems to try.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
(C) 2011 Copyright Math Open Reference. All rights reserved

