
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. 
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.

Manipulative 2: Parts of a regular hexagon. Click on the check boxes to see the parts of the hexagon. 
Illustration  Formula  Description 

n = 6  number of sides  
s  length of a side  
p = 6s  perimeter  
radius of the incircle  
r_{1} = 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. 
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


Table 2: How to construct a regular hexagon from its circumcircle. 
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 
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.  A truncated triangular tiling of hexagons. Click on the blue points in the manipulative and drag them to change the figure. 
Table 4: tessellations of a hexagon 
#  A  B  C  D 
E  F  G  H  I 
J  K  L  M  N 
O  P  Q  R  S 
T  U  V  W  X 
Y  Z 
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