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Question 90
If each interior angle of a regular polygon is $$\left(128\frac{4}{7}\right)^\circ$$ , then what is the sum of the number of its diagonals and the number of its sides?
Solution
Interior angle = 180 - $$\frac{360}{n}$$
$$128\frac{4}{7}^\circ =Â 180 - \frac{360}{n}$$
$$\frac{900}{7}^\circ = 180 - \frac{360}{n}$$
$$ \frac{360}{n} = 180 -Â \frac{900}{7}$$
$$ \frac{360}{n} =Â \frac{360}{7}$$
Side(n) = 7
Number of diagonals = $$\frac{n(n - 3)}{2} =Â \frac{7(7 - 3)}{2}$$
= $$\frac{28}{2}$$ = 14
Sum of the number of its diagonals and the number of its sides = 7 + 14 = 21
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