Question

Find the number of sides of a regular polygon when each of its angles has a measures of (i) ${160}^{\circ }$ (ii) ${135}^{\circ }$ (iii) ${175}^{\circ }$ (iv) ${162}^{\circ }$ (v) ${150}^{\circ }$.

Answer 1

In a n-sided regular polygon, each angle = $\frac{2n-4}{n}$ right angles or $[\frac{2n-4}{n}\times {90}^{\circ }]$
(i) When each interior angle = ${160}^{\circ }$
Let number of sides of regular polygon=n
$\therefore \frac{2n-4}{n}\times {90}^{\circ }={160}^{\circ }\frac{2n-4}{n}=\frac{{160}^{\circ }}{{90}^{\circ }}=\frac{16}{9}$
By cross multiplication:
$18n-36=16n\Rightarrow 18n-16n=36\Rightarrow 2n=36\Rightarrow n=\frac{36}{2}=18$
$\therefore$ Regular polygon is of 18 sided.
(ii) Each interior angle = ${135}^{\circ }$
Let number of sides of regular polygon = n
$\frac{2n-4}{n}\times {90}^{\circ }={135}^{\circ }\Rightarrow \frac{2n-4}{n}=\frac{135}{90}=\frac{3}{2}$
By cross multiplication:
$4n-8=3n\Rightarrow 4n-3n=8\Rightarrow n=8$
$\therefore$ The regular polygon has 8 sides.
(iii) Each interior angle = ${175}^{\circ }$
Let number of sides of regular polygon = n
$\therefore \frac{2n-4}{n}\times {90}^{\circ }={175}^{\circ }\Rightarrow \frac{2n-4}{n}=\frac{175}{90}=\frac{35}{18}$
By cross multiplication:
36n-72=35n
$\Rightarrow 36n-35n=72\Rightarrow n=72$
$\therefore$ Regular polygon is of 72 sides.
(iv) Each interior angle = ${162}^{\circ }$
Let number of sides of regular polygon=n
$\therefore \frac{2n-4}{n}\times {90}^{\circ }={162}^{\circ }$
$\Rightarrow \frac{2n-4}{n}=\frac{162}{90}=\frac{9}{5}$
By cross multiplication:
10n-20=9n
$\Rightarrow 10n-9n=20\Rightarrow n=20$
$\therefore$ Regular polygon is 20sides.
(v) Each interior angle = ${150}^{\circ }$
Let number of sides of regular polygon = n
$\therefore \frac{2n-4}{n}\times {90}^{\circ }={150}^{\circ }\Rightarrow \frac{2n-4}{n}=\frac{150}{90}=\frac{5}{3}$
By cross multiplication,
$6n-12=5n\Rightarrow 6n-5n=12\Rightarrow n=12$
$\therefore$ The regular polygon of 12 sides.