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?

Question

Philoid · Accepted Answer

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.