In geometry, any polygon with ten sides and ten angles and ten vertices is called as Decagon. The sum of angles in a decagon is 1440°.

In real life, Decagon shape can be seen in few coins, tiles and mugs. 2 Peso Coin of the Republic of the Philippines has a decagon 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 decagon are 35 since n = 10 in a decagon.

## Regular Decagon

A Decagon having all sides of equal length and equal angles is called regular decagon. The figure above shows a regular decagon.
Each of its internal angles is equal to 144° (The formula to find an interior angle of an n-sided polygon is given by $\frac{(n-2) 180}{n}$degrees). 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 decagon is 36°.

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 8 triangles in a regular decagon.

## Irregular Decagon

An Irregular Decagon has sides of different lengths.

Irregular polygons are not symmetric and so are not considered as having a centre. Thus have no central angle.

## Convex Decagon

Geometrically, in a figure if every line segment drawn between any two interior points lies entirely inside the figure is called a convex figure. Similarly convex Decagon is a 2D geometric figure in which every line segment drawn between any two interior points lies inside the decagon. Any regular Decagon is a Convex Decagon.

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 Decagon, the sum of angles is given by (10 - 2) × 180° = 8 × 180° = 1440°.
In a convex Decagon, the sum of the degree measures of the exterior angles, one at each vertex, is 360°.

## Concave Decagon

A concave Decagon is defined as a polygon with one or more interior angles greater than 180°. Also a Decagon that is not convex is a concave Decagon.

For example, A ten pointed star polygon is called a Concave Decagon.

In this 10-Star decagon, we can see that the interior angles at H, F, D, B, J 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 AI, IG, GE, EC and CA.

## Area of a Decagon Formula

The area of a Decagon can be found in different ways. We can use various formulas depending on the data when the decagon is regular.

Case (i) When the side length, s, is given the area of any n sided polygon is given by, Area = $\frac{s^2n}{4tan(\frac{180}{n})}$ where, the tangent function is always calculated in degrees.

Putting n = 10, we get the area of regular decagon with side length s as, Area = 7.6942 s2.

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 = a2n tan($\frac{180}{n}$) where, n is the number of sides of a polygon.

Putting n = 10, we get the area of regular decagon with Apothem a, Area = 3.2492 a2.

Case(iii) When an 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 = 10, we get the area of regular decagon with radius r, Area = 2.9389 r2.

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 Decagon, if the distance between any two parallel sides or the height when the decagon stands on one side as base is given, then the formula for area is given by, Area = 2.5ds where s is the side length.In case of an irregular Decagon, area can be found by breaking the decagon into 10 triangles and finding the area of each triangle and adding them up.

## Interior Angles of a Decagon

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 Decagon is equal to 144° (An interior angle of any polygon is given by $\frac{(180n–360)}{n}$. Here n=10 for a decagon).

## Exterior Angles of a Decagon

An exterior angle of a Decagon 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 decagon is 36° (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°.

## Decagon Examples

### Solved Examples

Question 1: Find the area of Decagon to nearest tenth with a side length of 12 cm.
Solution:

The length of each side of a decagon, s = 12 cm

Area of Decagon = 7.6942 s2

= 7.6492 X 12 X 12

= 1101.4848

= 1101.5 sq.cm.

Thus, the area of a decagon with side length as 12 cm is 1108.0sq.cm.

Question 2: Find the area of a regular Decagon with a radius of 6 cms to the nearest hundredth.
Solution:

The radius of a decagon,r = 6 cms

Area of Decagon  =($\frac{1}{2}$) r2nSin($\frac{360}{n}$)  = ($\frac{1}{2}$) r2(10)Sin($\frac{360}{10}$)  = 2.9389 r2

= 2.9389 x (6)2.

= 105.8013 sq.cm

Thus, the area of a decagon with radius of 6 cms is 105.80 sq.cms.