Question

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

Answer 1

Hint: We assume a regular heptagon with interior angle as $x$. We have the sum of the interior angles of a polygon with $n$ sides as $\left( n-2 \right)\times 180{}^\circ $ or $\left( n-2 \right)\pi $ radians. From this relation, we will calculate the sum of the angles of a heptagon and equate it to the sum of the interior angles of our assumed heptagon to get the value of the interior angle.

Complete step-by-step solution
Let the heptagon with interior angle $x$ be shown in the below figure.

Now the sum of the interior angles in the above heptagon is
$\begin{align}
  & S=x+x+x+x+x+x+x \\
 & S=7x....\left( \text{i} \right) \\
\end{align}$
But the sum of the interior angles of a heptagon having $7$sides in degrees is given by
$\begin{align}
  & S=\left( n-2 \right)180{}^\circ \\
 & \Rightarrow S=\left( 7-2 \right)180{}^\circ \\
 & \Rightarrow S=900{}^\circ \\
\end{align}$
From equation $\left( \text{i} \right)$ we can write
$\begin{align}
  & 7x=900{}^\circ \\
 & \Rightarrow x=\dfrac{900{}^\circ }{7} \\
 & \Rightarrow x=128{}^\circ 34'17'' \\
\end{align}$
Now the sum of interior angles of a heptagon having $7$sides in radians is given by
$\begin{align}
  & S=\left( n-2 \right)\pi \\
 & \Rightarrow S=\left( 7-2 \right)\pi \\
 & \Rightarrow S=5\pi \text{ rad} \\
\end{align}$
From equation $\left( \text{i} \right)$ we can write
$\begin{align}
  & 7x=5\pi \\
 & \Rightarrow x=\dfrac{5}{7}\pi \\
\end{align}$
$\therefore $ The interior angle of a heptagon is $128{}^\circ 34'17''$ or $\dfrac{5}{7}\pi $ radians.

Note: We can also directly use the formula for interior angle of a polygon with $n$ sides as $\left( 180{}^\circ -\dfrac{360{}^\circ }{n} \right)$, then the interior angle of an heptagon with $7$ sides is
$\begin{align}
  & x=\left( 180{}^\circ -\dfrac{360{}^\circ }{7} \right) \\
 & \Rightarrow x=\left( \dfrac{1260{}^\circ -360{}^\circ }{7} \right) \\
 & \Rightarrow x=\dfrac{900{}^\circ }{7} \\
 & \Rightarrow x=128{}^\circ 34'17'' \\
\end{align}$
We know that $180{}^\circ =\pi $ radians, then
$\begin{align}
  & 128{}^\circ 34'17''=\dfrac{\pi }{180{}^\circ }\times \dfrac{900{}^\circ }{7} \\
 & \Rightarrow 128{}^\circ 34'17''=\dfrac{5}{7}\pi \\
\end{align}$
From both the methods we got the same result.
We only use the above formula in case the polygon is regular, for irregular polygons we will use the formula $\left( n-2 \right)\times 180{}^\circ $ and then we will calculate the interior angles as shown in the problem. In some cases, they mention the values of one interior angle and give the relation between the remaining interior angles and ask to find all interior angles. Then the method we used in the problem is very useful.