## Elementary Geometry for College Students (6th Edition)

Given In a regular polygon, each interior angle measures 135°. If each side of the regular polygon measures 4.2 cm We need to find out the perimeter of the polygon For given polygon radius makes interior angle into two equal half angles. At any corner of polygon, the half angle = $\frac{135 ^{\circ}}{2}$ = 67.5 $^{\circ}$. The triangle formed by radius of polygon and apothem is a righr angle triangle. Half of the central angle = 180- (90 +67.5) = 22.5 $^{\circ}$ The full central angle= 22.5 $^{\circ}$ * 2 = 45$^{\circ}$. We know the relationship between the central angle and number of sides. Central angle = $\frac{360 ^{\circ} }{n}$ 45 $^{\circ}$ = $\frac{360 ^{\circ} }{n}$ n = $\frac{360}{45}$ = 8 Therefore the polygon consist of 8 sides The perimeter of polygon = np = 8*4.2 = 33.6 cm