Question

What is the total number of diagonals for a hexagon and a heptagon?

Answer 1

Hint: From the question given we have to find the total number of diagonals for a hexagon and a heptagon. in order to solve this question, we will use formula for number of diagonals of a polygon that is $\dfrac{n\left( n-3 \right)}{2}$ where n is number of vertices of the polygon. In this formula we will keep n=6 for hexagon and n=7 for heptagon. This way we will get our desired diagonals.

Complete step-by-step answer:
From the question given we have to find the total number of diagonals for a hexagon and a heptagon.
we will use formula for number of diagonals of a polygon that is
 $= \dfrac{n\left( n-3 \right)}{2}$
where n is the number of vertices of the polygon.
Now, we know that in a hexagon there are 6 vertices and we will find the number of diagonals by using the formula given below
$= \dfrac{n\left( n-3 \right)}{2}$
Now, n= 6,
Putting the value of n =6 in the above formula, we will find the number of diagonals of the hexagon
Therefore, number of diagonals of hexagon is
$= \dfrac{6\left( 6-3 \right)}{2}$
$= \dfrac{18}{2}$
$= 9$
Hence there are 9 diagonals for a hexagon.
Now, we know that in a heptagon there are 7 vertices and we will find the number of diagonals by using the formula given below
$= \dfrac{n\left( n-3 \right)}{2}$
Now, n= 7,
Putting the value of n =7 in the above formula, we will find the number of diagonals of the heptagon Therefore, number of diagonals of heptagon is
$= \dfrac{7\left( 7-3 \right)}{2}$
$= \dfrac{28}{2}$
$= 14$
Hence there are 14 diagonals for a heptagon.

Note: Whenever we face these types of questions the key concept is that we have to count the vertices of the polygon and put that count as n in the formula. Student should also know that the formula for sum of the interior angles of a polygon is $180\left( n-2 \right)\deg rees$. and also interior angles of a regular polygon is $\dfrac{\left( {{180}^{\circ }}\left( n \right)-{{360}^{\circ }} \right)}{n}\deg rees$, where n is the number of vertices of a polygon.