
Hexagon inscribed in a circle
This page shows how to construct (draw) a
regular hexagon
inscribed in a circle with a compass and straightedge or ruler. This is the largest hexagon that will fit in the circle, with each
vertex
touching the circle. In a regular hexagon, the side length is equal to the distance from the center to a vertex, so we use this fact to set the compass to the proper side length, then step around the circle marking off the vertices.
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
As can be seen in Definition of a Hexagon,
each side of a regular hexagon is equal to the distance from the center to any vertex.
This construction simply sets the compass width to that radius, and then steps that length off around the circle
to create the six vertices of the hexagon.
Proof
The image below is the final drawing from the above animation, but with the vertices labelled.

Argument 
Reason 
1 
A,B,C,D,E,F all lie on the circle center O 
By construction.

2 
AB = BC = CD = DE = EF 
They were all drawn with the same compass width. 
From (2) we see that five sides are equal in length, but the last side FA was not drawn with the compasses.
It was the "left over" space as we stepped around the circle and stopped at F.
So we have to prove it is congruent with the other five sides. 
3 
OAB is an equilateral triangle 
AB was drawn with compass width set to OA, and OA = OB (both radii of the circle). 
4 
m∠AOB = 60° 
All interior angles of an equilateral triangle are 60°. 
5 
m∠AOF = 60° 
As in (4) m∠BOC, m∠COD, m∠DOE, m∠EOF are all &60deg;
Since all the central angles add to 360°,
m∠AOF = 360  5(60) 
6 
Triangle BOA, AOF are congruent 
SAS See Test for congruence, sideangleside. 
7 
AF = AB 
CPCTC  Corresponding Parts of Congruent Triangles are Congruent 
So now we have all the pieces to prove the construction 
8 
ABCDEF is a regular hexagon inscribed in the given circle 
 From (1), all vertices lie on the circle
 From (20), (7), all sides are the same length
 The polygon has six sides.

 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
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