Two regular polygons have the number of their sides in the ratio 2:1 and their interior angle in the ratio 5:4. The number of sides of the two polygons are?
Answers
Let the regular polygon A have n sides and regular polygon B have 2n sides.
The sum of the interior angles of A is (n-2)*180 = 180n - 360. So each interior angle = (180n - 360)/n deg.
The sum of the interior angles of B is (2n-2)*180 = 360n - 360. So each interior angle = (360n - 360)/2n deg. Now the ratio of the interior angles of A and B
(180n - 360)/n:(360n - 360)/2n::2:3
(180n - 360)*3*2/n =(360n - 360)*2/n
(180n - 360)*3*2 =(360n - 360)*2
(180n - 360)*3 =(360n - 360), or
540n - 1080 = 360n - 360, or
180n = 1080 - 360 = 720, or
n = 720/180 = 4
A has 4 sides and B has 8 sides. The difference in the number of sides of A and B is 4.
Check: The interior angles of A, which is a square, is 90 deg.
The interior angles of B, which is an octagon, is 180-(360/8) = 180–45 = 135 deg.
HCF of 90 and 135 is 45 so ratio of interior angles of A and B is 90/45:135/45 = 2/3. Correct.