Question

Find the magnitude, in radians and degrees, of the interior angle of a regular duo-decagon.
\[{\text{A}}{\text{. }}\left( {\dfrac{{7\pi }}{6}} \right);{150^ \circ }\]
\[{\text{B}}{\text{. }}\left( {\dfrac{{5\pi }}{6}} \right);{150^ \circ }\]
\[
  {\text{C}}{\text{. }}\left( {\dfrac{{5\pi }}{6}} \right);{130^ \circ } \\
  {\text{D}}{\text{. None of these}} \\
 \]

Answer 1

Hint: A polygon is a closed figure which includes an equal number of sides and angles. For a closed figure, a minimum of 3 sides are required, and the polygon with three sides and three angles is known as a triangle. Similarly, the polygon with four sides is known as a quadrilateral. Duo-decagon is a polygon having twelve edges and twelve interior angles where the interior angle is referred to as the angle between adjacent sides in a polygon.
To find the sum of the interior angles of the polygon, we multiply the number of triangles in the polygon\[{180^ \circ }\]. The number of triangles in a polygon is found by using the formula\[(n - 2)\], where n is the number of sides of a polygon. For example, if the number of sides of a polygon is 6, then the number of triangles will be \[\left( {n - 2} \right) = \left( {6 - 2} \right) = 4\] triangles. Similarly, first, find the number of triangles for the duo-decagon. The sum of the interior angle of the polygon is \[(n - 2)\pi \], where each angle of a polygon is \[\dfrac{{(n - 2)\pi }}{n}\] where n is the number of sides.

Complete step by step answer:
The number of sides in duo-decagon is 12.
So, the number of triangles will be \[\left( {n - 2} \right) = (12 - 2)\]
We get each angle of duo-decagon in radians as:
\[
  \theta = \dfrac{{(12 - 2)\pi }}{{12}} \\
   = \dfrac{{10\pi }}{{12}} \\
   = \left( {\dfrac{{5\pi }}{6}} \right) \\
 \]
Now, to convert the angle from radian to the degree we put the value of \[\pi = {180^ \circ }\]. Hence we get
\[
  \theta = \left( {\dfrac{{5 \times {{180}^ \circ }}}{6}} \right) \\
   = {150^ \circ } \\
 \]
\[{\text{Answer is B}}{\text{. }}\left( {\dfrac{{5\pi }}{6}} \right);{150^ \circ }\]

Note: In general, before proceeding for finding the total interior angles of a polygon, we find the numbers of triangles in the polygon. Angles are generally measured in radians and degrees where\[1{\text{ radian}} = \dfrac{{180}}{\pi }{\text{ degrees}}\].