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.

Total number of diagonals=n(n-3)/2

Now, n(n-3)/2=54

n^2-3n=54×2

So n^2-3n=108

n^2-3n-108=0

By using quadratic formula,

$n = 3 + - \sqrt{}9 - 4 \times 1 - 108 \div 2 \\ = 3 + - \sqrt{}441 \div 2 \\ = 3 + - 21 \div 2 \\ n = 3 + 21 \div 2 \: or \: n = 3 - 21 \div 2 \\ n = 12 \: or \: n = - 9$

Here, n=12

Interior angle sum=(n-2)180

That is (12-2)180=1800

Each angle=1800/12

=150

HOPE IT HELPS YOU DEAR.......