Question

If in two circles, arcs of the same length subtend angles ${60}^{\circ }$ and ${75}^{\circ }$ at the centre, find the ratio of their radii.

Answer 1

Step 1: Convert degree into radian
Let the radii of the two circles be $r_1$ and $r_2$.
Let an arc of length 𝑙 subtends an angle of ${60}^{\circ }$ at the centre of the circle of radius $r_1$.
While let an arc of length 𝑙 subtends an angle of ${75}^{\circ }$ at the centre of the circle of radius $r_2$.
${\theta }_1={60}^{\circ }=\frac{\pi }{3}$ radian
And
${\theta }_2={75}^{\circ }=\frac{5\pi }{12}$ radian

Step 2: Use the relation between length of arc and angle subtended at the centre
We know that in a circle of radius 𝑟 unit, if an arc of length 𝑙 unit subtends an angle $\theta$ radian at the centre,
then $\theta =\frac{l}{r}orl=r\theta$
$\therefore l=\frac{r_1\pi }{3}$ and $l=\frac{r_2\left(5\pi \right)}{12}$

Step 3: Solve for $\frac{r_1}{r_2}$
$\Rightarrow \frac{r_1\pi }{3}=\frac{r_2\left(5\pi \right)}{12}\Rightarrow \frac{r_1}{r_2}=\frac{5}{4}$