Question

If a regular polygon has 54 diagonals, what is the measure of each interior angle of the polygon​

Answer 1

Answer:

The number of diagonals for a polygon having  n  sides is given by  n∗(n−3)/2  

You can easily derive it.

Diagonal is a line segment joining two edges of the polygon which are already not a polygon edge.

The number of edges from  n  sides will be  nC2 .

The number of edges will be  n .

Therefore, total number of diagonals =  nC2−n  

= n∗(n−3)/2  

Now,

n∗(n−3)/2=54  

=>n2−3n=108  

=>(n−12)(n+9)=0  

Solving this, you get

n=12

Interior angle sum = (n - 2 ) * 180

= ( 12 - 2 ) * 180

= 1800

each angle = $\frac{1800}{12}$ = 150