# Decagon

## Definition

A decagon is a polygon with ten sides and ten angles. The term ‘decagon’ is derived from the Latin word ‘decagonum’ where ‘deca’ means ten, and ‘gonium’ means ‘angles’. Thus ‘decagon’ means ten angles and thus also known as 10-gon.

The below-given properties and formulas to find the perimeter and area only apply to the regular decagon.

## Properties

1. Has 10 sides of equal length; in decagon ABCDEFGHIJ, AB = BC = CD = DE = EF = FG = GH = HI = IJ = JA
2. Has 10 interior angles each measuring 144°; so ∠ABC = ∠BCD = ∠CDE = ∠DEF = ∠EFG =∠FGH = ∠GHI = ∠HIJ = ∠IJA = ∠JAB
3. The sum of all nine interior angles is 1440°; so ∠ABC = ∠BCD + ∠CDE + ∠DEF + ∠EFG +∠FGH + ∠GHI + ∠HIJ + ∠IJA + ∠JAB = 1440°
4. Has 10 exterior angles, each measuring 36°
5. Has 35 diagonals, example AC, BD, CE, DG, EI, and FJ

## Formulas

### Perimeter

The formula for finding the perimeter of a decagon is given below:

Find the perimeter of a regular decagon with each side measuring 5.5 cm.

Solution:

As we know,
Perimeter (P) = 10a, here a = 5.5 cm
= 10 x 5.5 cm
= 55 cm

### Area

The formula for finding the area of a decagon is given below:

The above equation is approximately equal to

A = 7.694 a2 square units

Find the area of a regular decagon having each side measuring 8 cm.

Solution:

As we know,
Area (A) = 5/2 a2 √(5+2√5), here a = 8 cm
= 7.694 a2
= 7.694 (8)2
= 7.694 x 64
= 492.41 cm2

### Angles

#### Interior Angle

The angle formed inside the decagon at its corners when the line segments join in an end to end fashion.

Sum of Interior Angles

The total measure of all the interior angles combined in the decagon. The formula is given below:

Sum of the interior angles = (n-2) x 180°, here n = number of sides

In decagon ABCDEFGHIJ, n = 10

Thus,

Sum of the interior angles = (10 -2) x 180°

= 1440°

One Interior Angle

The measure of one interior angle can be obtained by dividing the sum of the interior angles by the number of sides in a decagon. The formula is given below:

One interior angle = (n-2) x 180°/n, here n = number of sides

In decagon ABCDEFGHIJ, n = 10

Thus,

Sum of the interior angles = (10-2) x 180°/10

= 144°

#### Exterior Angle

The angle formed by any side of the decagon and the extension of its adjacent side. The formula is given below:

Exterior angle = 360°/n, here n = number of sides

In decagon ABCDEFGHIJ, n = 10

Thus,

Exterior angle = 360°/10

= 36°

## Types

Depending on the sides, angles, and vertices, decagon shapes are classified into the following types:

1. Regular Decagon: Has ten sides of equal length and ten interior angles, each measuring 144°, and exterior angles of 36° each. It has ten lines of symmetry and ten reflectional symmetries. All regular decagon is convex.
2. Irregular Decagon: Does not have all sides equal or all interior angles equal, but the sum of all ten interior angles is equal to 1440°. An irregular decagon can be both convex and concave.
3. Convex Decagon: Have all ten vertices pointing outwards. No interior angle of a convex decagon measure more than 180°, and all the diagonals lie inside the closed figure. A convex decagon can be both regular and irregular.
4. Concave Decagon: Have at least one vertex pointing inwards with an interior angle greater than 180°. At least one diagonal lie outside the closed figure. Thus all concave decagon are irregular.

## Examples in Real Life

• The shape of some coins, mirrors, plates, umbrellas, drums, watches, cutlery, and coasters.
• A star or any star-shaped object is an irregular decagon.
• A quasiperiodic crystal or quasicrystal is composed of overlapping decagon clusters.
• Widely used while doing wall art.