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 P and Q is 45°.find the number of sides of p.
b. The area of a semi-circle is 32π \(cm^2\). Find, in terms of π, the circumference of the semi-circle.
Explanation
a. Let the number of sides of polygon Q be x
therefore, No. of sides of polygon P = 2x
then, \(\frac{360}{x} - \frac{360}{2x}\) = 45,
\(\frac{360}{x} - \frac{180}{x}\) = 45
L. C. M = x
360 - 180 = 45x
x = \(\frac{180}{4}\) = 4
therefore, the number of P polygon = 2x = 2 x 4 = 8
b. Area of semi-circle = \(\frac{\pi r^2}{2}\)
but Area of the semi-circle = 32π
\(\frac{\pi r^2}{2}\) = 32π
\(r^2 = 32\times2 = 64\)
\(r^2 = \sqrt{64}\) = 8cm.
Circumference of semi-circle = πr + 2r
= π(8) + 2(8)
= 8π + 16
= 8(π + 2 ) cm