Question

What is the formula to find the area of a regular dodecagon?

Answer 1

Hint: A closed figure that has at least three sides is known as a polygon. There are various kinds of polygons and a dodecagon is also a kind of polygon with twelve sides. A dodecagon can be regular or it can be irregular. A triangle, rectangle, pentagon, hexagon, heptagon, octagon, etc are all different kinds of polygons with a different number of sides.

Complete step-by-step solution:
Various kinds of polygons exist in geometry. A three-sided polygon is known as the triangle, a four-sided polygon is known as the rectangle, a pentagon is a five-sided polygon, and so on. A polygon has two types, it can be a regular polygon or an irregular polygon. A regular polygon has equal sides and equal angles but in an irregular polygon neither the sides are equal nor the angles are equal. The sum of all exterior angles of a polygon comes out to be \[360{}^\circ \].
A polygon with twelve sides, twelve angles, and twelve vertices is known as a dodecagon. This type of polygon can be of any type. Depending on its properties it can be a regular polygon, irregular, concave, and convex. The sum of all the interior angles in a regular dodecagon comes out to be \[1800{}^\circ \] and each angle of a regular dodecagon is \[150{}^\circ \]. A dodecagon has \[54\] diagonals in it.
A regular dodecagon is a polygon that has all the sides of equal length and all the vertices are equidistant from the center of the polygon. Also, this polygon is symmetrical in shape.
An irregular dodecagon has different sides and angles are also different. All the irregular dodecagons have an infinite number of variations and all of them have twelve sides but they still look different from each other.
The area of a regular dodecagon can be obtained by the below-given formula.
\[A=3\times (2+\sqrt{3})\times {{s}^{2}}\]
Where ‘A’ is known as the area of the regular dodecagon and ‘s’ is the length of the sides of a dodecagon.

Note: A dodecagon can be broken into a series of triangles with the help of diagonals which are drawn from one vertex to another one. The perimeter of a dodecagon can be obtained by adding all its sides or if the dodecagon is regular then the perimeter can be obtained by multiplying the length of one side with the total number of sides.