Question

Find the magnitude in radians and degrees of the interior angle of a regular heptagon.

Answer 1

Hint: We know that an interior angle of a regular heptagon is located within the boundary of a heptagon. Also the sum of all interior angles of a polygon can be found using the formula \[S=(n-2)\times 180\] degree where ‘n’ is number of sides of the polygon and to calculate each angle of the regular polygon we will get it by dividing the sum by number of sides.
\[\Rightarrow \]Each angle \[=\left( \dfrac{n-2}{n} \right)\times 180\] degree.

Complete step-by-step answer:
We have been asked the magnitude in radians and degrees of the interior angle of a regular heptagon.
Now as we know that a regular heptagon means a polygon with 7 sides and 7 angles and each of the sides and angles are equal to each other.
\[\Rightarrow \]For a regular heptagon ‘n’ = 7
We know that the measure of each interior angle of a regular polygon having ‘n’ sides is given by as follows:
Each angle \[=\left( \dfrac{n-2}{n} \right)\times 180\] degree
So for a regular octagon, each angle \[=\left( \dfrac{7-2}{7} \right)\times 180\]degree
\[=\dfrac{5}{7}\times 180\] degree
\[=\dfrac{900}{7}\] degree
We know that 1 degree \[=\dfrac{\pi }{180}\] radians
\[\Rightarrow \left( \dfrac{900}{7} \right)=\dfrac{\pi }{180}\times \dfrac{900}{7}\] radians
\[=\dfrac{5\pi }{7}\] radians
Therefore, the magnitude of interior angles of a regular heptagon in radians and degrees are radians \[\dfrac{5\pi }{7}\] and \[\dfrac{900}{7}\] degree respectively.

Note: Be careful while conversion of degree into radians and use the formula 1 degree = \[\dfrac{\pi }{180}\] radians and don’t use 1 degree = \[\dfrac{180}{\pi }\]radians in hurry. Also remember that a regular polygon means each side and angle of the polygon are equal to each other. Also, be careful while calculating each angle of the polygon and use the formula of each angle \[=\left( \dfrac{n-2}{n} \right)\times 180\] degree. Don’t miss that the sum \[\left( n-2 \right)\times 180\] is divided by ‘n’ number of sides of the polygon.