In geometry, any polygon with six sides and six angles and six vertices is called as Hexagon. The sum of angles in a Hexagon is 720°.
Hexagon Picture
In real life, Hexagon shape can be seen in ice crystals by which snow is formed. Some snowflakes have the shape of regular hexagons. The cells of beehive seen in honeycombs are also hexagonal in shape. The number of diagonals in a polygon is given by the formula $\frac{n(n-3)}{2}$. Hence the number of diagonals that can be drawn in a Hexagon are 9 since n = 6 in a Hexagon.

A Hexagon having all sides of equal length and equal angles is called regular Hexagon.

Regular Hexagon

The figure above shows a regular Hexagon. Each of its internal angles is equal to 120°. The formula to find an interior angle of an n-sided polygon is given by $\frac{[(n-2) \times 180]}{n}$. In a polygon, the central angle is the angle made at the centre of the polygon by any two adjacent vertices and is given by the formula $\frac{360^o}{n}$. Thus the central angle measure of a regular hexagon is 60 °.

The number of triangles that can be drawn by drawing the diagonals from a given vertex in general is given by n - 2 and so we can draw 4 triangles in a regular hexagon.
An Irregular Hexagon has sides of different lengths.

Irregular Polygon

Irregular polygons are not symmetric and so are not considered as having a centre. Thus have no central angle.
Geometrically, in a figure if any line segment drawn between any two interior points lies entirely inside the figure, then it is called a convex figure. Similarly convex Hexagon is a 2D geometric figure in which any line segment drawn between any two interior points lies inside the Hexagon. Any regular Hexagon is a Convex Hexagon.

In a convex polygon with n sides, the sum of interior angles is given by the following equation: S = (n -2) × 180°.

So, in a convex Hexagon, the sum of angles is given by (6 -2) × 180° = 4 × 180° = 720°.
In a convex Hexagon, the sum of the degree measures of the exterior angles, one at each vertex, is 360°.
A concave polygon has one or more vertices which point towards the interior and looks like “pushed in”. A concave Hexagon is defined as a polygon with one or more interior angles greater than 180°. Also a Hexagon that is not convex is a concave Hexagon.
For example, A three pointed star polygon is called a Concave Hexagon.

Concave Hexagon

In this 3-Star hexagon, we can see that the interior angles at B, D, F are greater than 180°. On observing this figure we can see that in a concave polygon some diagonals will lie outside the polygon, like diagonal AC, CE and EA.
The area of a Hexagon can be found in different ways. We can use various formulas depending on the data when the hexagon is regular.

Case (i) When the side length, s, is given the area of any n sided polygon is given by,
Area = $\frac{s^{2}n}{4tan}$($\frac{180}{n}$), where the tangent function is always calculated in degrees.
Putting n = 6, we get the area of regular hexagon with side length s as, Area = 2.5981 s2.

Area of a Hexagonal Formula

Case (ii) When an Apothem ‘a’(Apothem of a regular polygon is a perpendicular line segment drawn from the center to the midpoint of
one of its sides) is given then the area of a polygon is given by :
Area = a2ntan($\frac{180}{n}$) where n is the number of sides of a polygon.
Putting n = 6, we get the area of regular hexagon with Apothem ‘a’, Area = 3.4641 a2.

Area of Hexagonal Formula

Case(iii) When a Radius ‘r’(Radius of a regular polygon is a distance from the center to any of the vertex) is given, then the area of a polygon is given by
Area =($\frac{1}{2}$) r2nSin($\frac{360}{n}$) where n is the number of sides of a polygon.
Putting n = 6, we get the area of regular hexagon with radius r, Area = 2.5981 r2.

Hexagonal Formula Area
If r is the radius and A is the apothem, the area is also given by Area = ½ r A
Case(iv) Alternatively in a regular Hexagon, if the distance between any two parallel sides or the height when the hexagon stands on one side as base is given as d, then the formula for area is given by:

Area = 1.5ds where s is the side length.

In case of an irregular Hexagon, area can be found by breaking the hexagon into 6 triangles and finding the area of each triangle and adding them up.
In a polygon, the interior angles always add up to a constant value and depends only on the number of sides no matter if it regular or irregular, convex or concave, or what size and shape it is. Each interior angle of any regular Hexagon is equal to 120° (An interior angle of any polygon is given by $\frac{(180n–360)}{n}$. Here n=6 for a hexagon).
Interior Angles of Hexagon
An exterior angle of a Hexagon is formed by extending any one of its sides. The non-straight angle adjacent to an interior angle is the exterior angle. Each exterior angle of any regular hexagon is 60° (In a polygon, an exterior angle forms a linear pair with its interior angle, so in general the exterior angle is given by 180-interior angle).
In a convex polygon, the sum of the exterior angles is 360°.
Exterior Angles of Hexagon

Solved Examples

Question 1: Find the area of Hexagon to nearest tenth with a side length of 10 cms.
The length of each side of a hexagon, s = 10 cms
   Area of Hexagon = $\frac{s^{2}n}{4tan(\frac{180}{n})}$ = $\frac{s^{2}6}{4tan(\frac{180}{6})}$ =2.5981 s2
                                  = 2.5981 X 10 X 10
                                  = 259.81
                                  = 259.81 sq.cms.
Thus the area of a hexagon with side length as 10 cms is 259.81 sq.cms.

Question 2: Find the area of a regular Hexagon with a radius of 6cms to the nearest hundredth.
The radius of a hexagon, r = 6 cms
      Area of hexagon  =($\frac{1}{2}$) r2nSin($\frac{360}{n}$)  = ($\frac{1}{2}$) r2(6)Sin($\frac{360}{6}$)  = 2.5981 r2
                                   = 2.5981 x (6)2.
                                   = 93.53
Thus the area of a hexagon with radius of 6 cms is 93.53