Sometime throughout the mid 1600s, a mathematician feeling the need for a descriptive word for a 12-sided polygon. Words dodecagon originates from the Greek, dōdeka-, which means 12, and also + -gōnon, which way angled. Dodecagons have the right to be regular, an interpretation all internal angles and also sides are equal in measure. Lock can likewise be irregular, v varying angles and also sides of different measurements. If friend ever try to attract a dodecagon freehand, you will no doubt do an irregular dodecagon.

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## Polygon

A dodecagon is a type of polygon with these properties:

It has actually two dimensionsIt has actually 12 right sides enclosing a spaceIt has 12 interior angles

## Finding Angles and also Perimeter that a consistent Dodecagon

No matter the shape, a constant polygon have the right to have that is exterior angles add to no an ext than 360°. Think: come go around the shape, you do a finish circle: 360°.

So, division 360° by the dodecagon"s twelve exterior angles. Each exterior angle is 30°.

That to be the basic part. The interior angles of a dodecagon are a bit harder. You deserve to use this generic formula to discover the amount of the internal angles for an n-sided polygon (regular or irregular):

Sum of internal angles = (n-2) x 180°Sum of interior angles = 10 x 180° = 1800°

Once you recognize the sum, you deserve to divide that by 12 to acquire the measure of each internal angle:

1800°/12 = 150°

This means each side intersects the next side only 30° much less than a straight line! That is one of two reasons illustration a consistent dodecagon freehand is for this reason difficult. The other reason is the difficulty of drawing 12 equal-length sides.

To calculation the perimeter the a regular dodecagon, multiply one side"s measurement, s, times 12:

Perimeter = 12 x sLength the one side: 17 mmPerimeter: 12 x 17 mm = 204 mm

## Area that a constant Dodecagon

The simple calculations room behind us. Currently let"s tackle area of a regular dodecagon. Because that a constant dodecagon v sides s, the area formula is:

A = 3 × (s)^2 × (2 + √3)

As one example, the 2017 brother one-pound coin is a regular dodecagon. One next of this beautiful coin is 6.278 mm in length. What is the area that this coin?

A = 3 × (s)^2 × (2 + √3)A = 3 × (6.278 mm)^2 × (2 + √3)A = 118.239852 mm^2 x 3.73205080757A = 441.277135143 mm^2A = 4.41277 cm^2

You will certainly seldom need that level that precision with your decimal places, therefore feel free to round as you wish; 441.277 mm^2 is very precise.

## Working through Dodecagons

You can attract a continuous dodecagon with only a pencil, paper, compass and straightedge, yet the measures are a little bit involved.

You will find few if any examples of normally occuring dodecagons in the organic world, yet coin minters choose the shape. That is very hard come counterfeit. One digital coin brochure lists some 449 dodecagonal coins of countless different nations.

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Besides the brother coin, Australia, Fiji and the Solomon archipelago mint dodecagonal coins.

## Try It!

Here is a regular dodecagon v sides measure 74 cm. What is its perimeter and also area?

Think prior to you peek!

### Perimeter:

Perimeter = 12 x sLength the one side: 74 cmPerimeter: 12 x 74 centimeter = 888 cm

### Area:

A = 3 × (s)^2 × (2 + √3)A = 3 × (74 cm)^2 × (2 + √3)A = 16,428 cm^2 x 3.73205080757A = 61,310.13065192 cm^2A = 6.13101307 m^2

Decagon