The ratio of the inferior to the exterior angles of a regular polygon is [tex]$x: y$[/tex].
(a) Find the number of sides of the polygon in terms of [tex]$x$[/tex] and [tex]$y$[/tex].
(b) Find the number of sides of the polygon if [tex]$x: y=3: 2$[/tex]
Answer (1)
Express the interior angle x and the exterior angle y in terms of the number of sides n of the polygon.
Find the ratio y x in terms of n : y x = 2 n − 2 .
Solve for n in terms of y x : n = 2 y x + 2 .
Substitute y x = 2 3 to find n = 2 ( 2 3 ) + 2 = 5 .
The number of sides of the polygon is $\boxed{5}.
Explanation
Define variables and state formulas. Let n be the number of sides of the regular polygon. The measure of each interior angle is given by x = n ( n − 2 ) 18 0 ∘ , and the measure of each exterior angle is given by y = n 36 0 ∘ . The ratio of the interior angle to the exterior angle is y x .
Express the ratio x/y in terms of n. We have the ratio y x = n 360 n ( n − 2 ) 180 = 360 ( n − 2 ) 180 = 2 n − 2 . We want to express n in terms of y x .
Solve for n in terms of x/y. Solving for n in terms of y x , we have: y x = 2 n − 2 2 y x = n − 2 n = 2 y x + 2 This answers part (a).
Find n when x/y = 3/2. Now, we are given that y x = 2 3 . Substituting this into the equation for n , we get: n = 2 ( 2 3 ) + 2 = 3 + 2 = 5 Therefore, the number of sides of the polygon is 5.
Examples
Understanding the relationship between interior and exterior angles of polygons is useful in architecture and design. For example, when designing a building with a regular polygonal base, architects need to calculate the angles to ensure structural integrity and aesthetic appeal. Knowing the ratio of interior to exterior angles helps in determining the optimal number of sides for the polygon, balancing space utilization and visual harmony.
