Hexagon
Pronunciation: /ˈhɛk.səˌgɒn/ Explain
 Figure 1: hexagons. 

A hexagon is a six sided
polygon.
The sides of a hexagon are straight line segments. A hexagon is a planar
figure, a figure that exists in a
plane.
A hexagon can be
concave
or
convex.
If a convex hexagon is
equilateral
(the sides are the same length), then the hexagon is a regular hexagon.

Article Index
Hexagon
Regular Hexagon
Parts of a Regular Hexagon
Formulas for a Regular Hexagon
How to Construct a Regular Hexagon From its Circumcircle
How to Construct the Center, Incircle and Circumcircle of a Regular Hexagon
Tessellations of a Regular Hexagon
Natural and Manufactured Hexagons
Regular hexagon
A regular hexagon is a six sided,
equilateral,
convex
polygon. An equilateral polygon has sides that are all the same length.
No line segment drawn between
any two points in a convex figure leaves the figure.
Click on the blue points and drag them to change the figure.
When is the length of a side different from the length of the circumradius?

Manipulative 1  Regular Hexagon Created with GeoGebra. 
Parts of a regular hexagon
 side or edge: One of the
line segment
that makes up hexagon. The length of a side of a regular hexagon
will be denoted s in this article.
 vertex:
Where two of the sides of the hexagon meet.
 interior:
The area inside the hexagon.
 exterior:
The area outside the hexagon.
 boundary:
The sides and vertices of the hexagon.
 center:
The point at the middle of the hexagon. The center of a regular hexagon
can be found at the intersection of the perpendicular bisectors of any
two sides. See How to Construct the Center,
Incircle and Circumcircle of a Regular Hexagon. The center of a
regular hexagon is also its centroid.
 incircle:
A circle whose center is at the center of a regular hexagon that touches
each side of the hexagon once. In this article, the radius of the
incircle will be denoted r_{1}.
 circumcircle:
A circle whose center is at the center of a regular hexagon that
intersects each of the vertices of the regular hexagon. In this article,
the radius of the circumcircle will be denoted r_{2}.
 apothem:
A line segment from the center of a regular hexagon to the midpoint
of one of the sides. The length of the apothem is the same as the radius
of the incircle and is denoted (r_{1}).
 sagitta:
A line segment from the center of a side to the edge of the
circumcircle of a regular hexagon that is collinear with the apothem.
 central angle:: The angle between two line segments extending
from the center of the regular hexagon to two adjacent vertices. The
central angle of a regular hexagon is always 72°.
 interior angle:
The angle between two adjacent sides. The interior angles of a regular
hexagon always measure 120°. The sum of the measures of interior angles
of any convex hexagon is 720°.
 exterior angle:
The angle between the extended side of a regular hexagon and an
adjacent side. The exterior angles of a regular hexagon always
measure 60°.
Illustration  Formula  Description 

n = 6  number of sides 
s  length of a side 
p = 6s  perimeter 
 radius of the incircle 
r_{2} = s  radius of the circumcircle 
 area of the hexagon 
 length of the apothem 
 length of the sagitta 
α = 72°  measure of the central angles 
β = 120°  measure of the interior angles 
γ = 60°  measure of the exterior angles 
Table 1: Formulas for a regular hexagon. 
How to Construct a Regular Hexagon from its Circumcircle
Step  Illustration  Description and Justification 
1 

Draw a circle. Label the center of the circle a. This is given. 
2 

Draw a circle with the center on the circumference of circle a
that is the same radius as circle a. Label the center of this circle
b. 
3 

Label the intersections of the circles c and d. 
4 

Draw line segments ab, ac, ad, bc
and bd. Note that all of these line segments are radii of congruent
circles. This means that they are all the same length. So triangles
Δabc and Δabd are equilateral triangles,
triangles whose sides have the same measure. The angles of equilateral triangles
also have the same measure. Therefore, triangles Δabc
and Δabd are congruent by the
SAS Congruence Theorem. 
5 

Draw another circle with the same radius as a with a center at
c. Label the new intersection of circle c and circle
a as e. 
6 

Draw another circle with the same radius as a with a center at
e. Label the new intersection of circle e and circle
a as f. Note that by the same arguments that were used in
step 5, triangles Δabc and Δace are
congruent equilateral triangles. 
7 

Continue the pattern until circles with centers at b, c,
d, e, f and g are drawn. We know this
is a regular hexagon because
 There are six sides. Count them.
 Each of the sides is a straight line segment. They were drawn as a straight
line segment.
 The sides are all the same size. See the reasoning in step 4.
 The figure is a convex.
Items 1 and 2 match the definition of a hexagon, so the figure must be a
hexagon. Items 1 through 4 match the definition of a regular hexagon, so the
figure must be a regular hexagon. 
Table 2: How to construct a
regular hexagon from its circumcircle. 
How to Construct the Center, Incircle
and Circumcircle of a Regular Hexagon
Step  Illustration  Discussion and Justification 
1 

The
center
of a regular hexagon is at the
point of concurrency
of perpendicular bisectors
of any two sides that are not opposite each other. 
2 

Draw the perpendicular bisector of any side. 
3 

Draw the perpendicular bisector of any other side that is not
opposite the side you used in step 2. 
4 

Label the intersection of the two perpendicular bisectors as 'center'. 
5 

Draw a circle with a center at the point labeled 'center' and the
edge at a point where a perpendicular bisector intersects a side. This
is the incircle. 
6 

Draw a circle with a center at the point labeled 'center' and
the edge at any vertex. This is the circumcircle. 
Table 3  How to construct the
center, incircle and circumcircle of a regular hexagon 
Tessellations of a Hexagon
A tessellation
of one or more polygon is an arrangement of those polygons that
fills a plane. The are a number of tessellations that use hexagons.
Most of the tessellations shown here use regular hexagons.


A regular tessellation of a hexagon. Three hexagons are around
each vertex. 
A trihexagonal tiling. Each vertex has a regular hexagon, an
equilateral triangle, a regular hexagon and an equilateral triangle. 
Table 4: tessellations of a hexagon 
Natural and Manufactured Hexagons



A hexagonal pavement. 
Hexagonal cloud formation on Saturn. 
A hexagonal mirror array. 



Carbon 36 fullerene molecules. 
Hexagonal basalt formation polished by a glacier. 
A hexagonal honeycomb. 
Table 5: Natural and Manufactured hexagons. Click on each image for more information on the image. 
References
 McAdams, David E.. All Math Words Dictionary, hexagon. 2nd Classroom edition 201501084799968. pg 90. Life is a Story Problem LLC. January 8, 2015. Buy the book
More Information
 McAdams, David E.. Polygon. allmathwords.org. Life is a Story Problem LLC. 10/22/2010.
Cite this article as:
McAdams, David E. Hexagon. 4/22/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/h/hexagon.html.
Image Credits
Revision History
4/22/2019: Update equations and expressions to new format. (
McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (
McAdams, David E.)
7/16/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (
McAdams, David E.)
10/22/2010: Initial version. (
McAdams, David E.)