If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2r, then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle. Is this statement false? Why?

True.

Let P and Q be two circles with radius r and 2r respectively. Let C1 and C2 be the centers of the circles P and Q respectively.


Let AB be the arc length of P and CD be the arc length of Q


Let θ1 and θ2 be the angle subtended by the arc AB and CD respectively on the center.


Given that AB = CD = l (say)


Now, arc length



CD = = l


∴ from the above two equations, we get



⇒ θ1 = 2θ2


∴ Angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle.


8