Question

The total measurement of all the interior angles of a regular polygon is equal to 10 right angles. Find out which kind of polygon it is.
(1) Decagon
(2) Hexagon
(3) Pentagon
(4) Heptagon

Answer 1

Hint: The total measurement of all the interior angles of an n-sided regular polygon is dependent on n. we use the formula of interior angles to find out the total interior angles of that n-sided polygon, which is a linear function of n. Then, we equate the function with 10 right-angles. We get a linear equation of n. We solve the equation to get the value of n which is the number of sides of that polygon.

Complete step-by-step solution:
Let the polygon be of n-sided. It has n interior angles.
We know that the sum of all the interior angles of an n-sided regular polygon is $\left( n-2 \right)\pi $.
So, each interior angle becomes $\dfrac{\left( n-2 \right)\pi }{n}$.
We also know that the sum of all exterior angles is $2\pi $ which is independent of the side number of a polygon.
It’s given that the total measurement of all the interior angles of the n-sided regular polygon is equal to 10 right angles.
10 right angles are equal to $10\times \dfrac{\pi }{2}=5\pi $.
Now, we equate the linear function of n with $5\pi $.
So, $\left( n-2 \right)\pi =5\pi $.
We solve the linear equation to get the value of n.
$\begin{align}
  & \left( n-2 \right)\pi =5\pi \\
 & \Rightarrow \left( n-2 \right)=5 \\
 & \Rightarrow n=5+2=7 \\
\end{align}$
So, the polygon is 7-sided. It’s a heptagon.
The correct option is (4).

Note: Although we mentioned the exterior angle measurement, it’s not required for the solution of the given problem. We also need to remember that the sum of all the interior and exterior angles of the polygon of n sides is $n\pi $. It helps to remember the formula of the sum of all interior angles.
We can check that for a heptagon the total interior angles will be $\left( n-2 \right)\pi $.
We put $n=7$. So, $\left( n-2 \right)\pi =\left( 7-2 \right)\pi =5\pi ={{900}^{\circ }}$.
The sum of the angles is ${{900}^{\circ }}$ which is equal to 10 right angles.