Question

One interior angle in a regular polygon is 162° how many sides does the polygon have?

Answer 1

20

Explanation:

Let n be the number of sides in the polygon. The sum of the interior angles in any n-sided polygon is 180(n-2) degrees. Since each angle in the given polygon measures 162, the sum of the interior angles of this polygon is also 162n. Therefore, we must have180(n-2) = 162n. Expanding the left side gives 180n - 360 = 162n, so 18n = 360 and n = 20.

We might also have noted that each exterior angle of the given polygon measures 180 - 162 = 18. The exterior angles of a polygon sum to 360, so there must be 360/18 = 20 of them in the polygon.

Answer 2

`\bf \textit{sum of all interior angles in a regular polygon}\\\\ S=180(n-2)\implies n\theta =180(n-2)\quad \begin{cases} n=\textit{number of sides}\\ \theta =\textit{interior angle}\\ ----------\\ \theta =162 \end{cases} \\\\\\ n(162) =180(n-2)\implies 162n=180n-360 \\\\\\ 360=18n\implies \cfrac{360}{18}=n\implies 20=n`