Question

Find the radian measure of the interior angle of a regular (i) Pentagon (ii) Hexagon (iii) Octagon

Answer 1

Hint: Degree and radian are related as \[Radian = \left( {\dfrac{\pi }{{{{180}^ \circ }}}} \right) \times Degree\]
In this question first find each angle of the given polygons in degree and then convert those angles in radian.
Each interior angle of a polygon can be determined by using the formula Interior angle \[\theta = \left( {n - 2} \right) \times \dfrac{{{{180}^ \circ }}}{n}\] , and then convert those angles in radian.

Complete step-by-step answer:
First we will find each interior angle of the polygons and them we will convert those angles in radian
Pentagon- A pentagon is a five sided polygon
Now since pentagon has 5 sides hence we can write \[n = 5\]
Now we know interior angle of a polygon can be determined by formula \[\theta = \left( {n - 2} \right) \times \dfrac{{{{180}^ \circ }}}{n}\] and in pentagon \[n = 5\] , hence by substituting the values we can write
  \[
  \theta = \left( {n - 2} \right) \times \dfrac{{{{180}^ \circ }}}{n} \\
   = \left( {5 - 2} \right) \times \dfrac{{{{180}^ \circ }}}{5} \\
   = 3 \times 36 \\
   = {108^ \circ } \\
 \]
So each internal angle is \[ = {108^ \circ }\]
Now we can write this angle in radian as
 \[
  Radian = \left( {\dfrac{\pi }{{{{180}^ \circ }}}} \right) \times Degree \\
   = \left( {\dfrac{\pi }{{{{180}^ \circ }}}} \right) \times 108 \\
   = \dfrac{{3\pi }}{5} \\
 \]
Hexagon- A hexagon is a six sided polygon
Hence we can write \[n = 6\]
Now we know interior angle of a polygon can be determined by formula \[\theta = \left( {n - 2} \right) \times \dfrac{{{{180}^ \circ }}}{n}\] and in hexagon \[n = 6\] , hence we can write
  \[
  \theta = \left( {n - 2} \right) \times \dfrac{{{{180}^ \circ }}}{n} \\
   = \left( {6 - 2} \right) \times \dfrac{{{{180}^ \circ }}}{6} \\
   = 4 \times 30 \\
   = {120^ \circ } \\
 \]
So each internal angle is \[ = {120^ \circ }\]
Now we can write this angle in radian as
 \[
  Radian = \left( {\dfrac{\pi }{{{{180}^ \circ }}}} \right) \times Degree \\
   = \left( {\dfrac{\pi }{{{{180}^ \circ }}}} \right) \times 120 \\
   = \dfrac{{2\pi }}{3} \\
 \]
Octagon- AN Octagon is a eight sided polygon
Hence we can write \[n = 8\]
Now we know interior angle of a polygon can be determined by formula \[\theta = \left( {n - 2} \right) \times \dfrac{{{{180}^ \circ }}}{n}\] and in hexagon \[n = 8\] , hence we can write
  \[
  \theta = \left( {n - 2} \right) \times \dfrac{{{{180}^ \circ }}}{n} \\
   = \left( {8 - 2} \right) \times \dfrac{{{{180}^ \circ }}}{8} \\
   = 6 \times \dfrac{{45}}{2} \\
   = {135^ \circ } \\
 \]
So each internal angle is \[ = {135^ \circ }\]
Now we can write this angle in radian as
 \[
  Radian = \left( {\dfrac{\pi }{{{{180}^ \circ }}}} \right) \times Degree \\
   = \left( {\dfrac{\pi }{{{{180}^ \circ }}}} \right) \times 135 \\
   = \dfrac{{3\pi }}{4} \\
 \]
Therefore the radian measure of the interior angle of a regular
I.Pentagon \[ = \dfrac{{3\pi }}{5}\]
II.Hexagon \[ = \dfrac{{2\pi }}{3}\]
III.Octagon \[ = \dfrac{{3\pi }}{4}\]

Note: Another method to find the each interior angle of a polygon is to find the sum of all internal angles of the polygon and then dividing them by the number of internal angles. Sum of internal angle of polygon is \[ = \left( {n - 2} \right) \times {180^ \circ }\]
To check: Sum of internal angle of hexagon with 6 sides is \[ = \left( {6 - 2} \right) \times {180^ \circ } = 4 \times {180^ \circ } = {720^ \circ }\]
Hence each internal angle will be \[ = \dfrac{{{{720}^ \circ }}}{6} = {120^ \circ }\] .