Theory question 111
Angles and Polygons(a) Two regular polygons P and Q are such that the number of sides of P is twice the number of sides of Q. The difference between the exterior angle of Q and P is 45°. Find the number of sides of P.
(b) The area of a semi-circle is 32π cm2. Find, in terms of π, the circumference of the semi-circle.
Show the model answer
Model answer
(a) Let Q have n sides, P 2n sides: 360/n − 360/(2n) = 45 ⇒ 180/n = 45 ⇒ n = 4, so P has 8 sides
(b) ½πr² = 32π ⇒ r = 8 cm
Curved part = πr = 8π cm; with the diameter, the perimeter = (8π + 16) cm
Explanation
A polygon with more sides has a smaller exterior angle.