# Dodecagon

‘Dōdeka’ – 12, and ‘gōnon’ – angle. So a dodecagon is a 2-dimensional shape with 12 sides that are straight lines. So, it has 12 angles.

## Properties

### Interior Angles

Each interior angle is 150°.

So, how many degrees are there in a dodecagon?

It simply means the sum of interior angles of a dodecagon, which is 12 × 150° = 1800°.

### Exterior Angles

Each exterior angle is 30°.

### Diagonals

The number of diagonals we can draw in a dodecagon is:

${\dfrac{1}{2}\times n\times \left( n-3\right)}$, here n = number of sides

${=\dfrac{1}{2}\times 12\times \left( 12-3\right)}$

= 54

The number of triangles = n – 2, here n = 12

= 12 – 2

= 10

### Lines of Symmetry

It has 12 lines of symmetry. 6 lines of symmetry go through the opposite vertices and 6 more lines pass through the mid points of the opposite sides as shown in the diagram.

## Types

### Regular and Irregular

1. Regular Dodecagon – It is a dodecagon with 12 equal sides and 12 interior angles measuring 150° each. It is symmetrical.

2. Irregular Dodecagon – It is a dodecagon with 12 sides and 12 interior angles having various measures.

### Convex and Concave

1. Convex Dodecagon – It is a dodecagon where no interior angle is greater than 180°. So, no vertex points inwards.

2. Concave Dodecagon – It is a dodecagon where at least one interior angle is greater than 180°. Some of its vertices can point inwards.

## Formulas

### Area

Find the area of a dodecagon with side of 5 cm.

Solution:

As we know,
Area ${\left( A\right) =3\times \left( 2+\sqrt{3}\right) \times s^{2}}$, here s = 5 cm
${\therefore A=3\times \left( 2+\sqrt{3}\right) \times 5^{2}}$
= 192.36 cm2

### Perimeter

Find the perimeter of a dodecagon with side of 6 cm.

Solution:

As we know,
Perimeter (P) = 12 × s, here s = 6 cm
∴P = 12 × 6
= 72 cm

Finding the side length of a dodecagon when AREA is known

Find the side length of a dodecagon whose area is 279 cm2.

Solution:

As we know,
Area ${\left( A\right) =3\times \left( 2+\sqrt{3}\right) \times s^{2}}$
${\therefore s=\sqrt{\dfrac{A}{3\left( 2+\sqrt{3}\right) }}}$, here A = 279 cm2
${ =\sqrt{\dfrac{279}{3\left( 2+\sqrt{3}\right) }}}$
= 4.99 cm