Question

Given

$r=\frac{1}{\pi }$ units
In the above figure,
Length of arc AB = 1 unit

• A. True
• B. False

Answer 1

The correct option is B

False

Applying the formula $L=r\theta$ to find the length of an arc, $\theta$ should be in radian. Here $\theta ={\pi }^{\circ })$ is given in degree. We will first convert to radian and use the relation $L=r\theta$.

$1^{\circ }=\frac{\pi }{180}radian$

${\pi }^{\circ }=\pi \times \frac{\pi }{{180}^o}radian$

$=\frac{{\pi }^2}{180}radian$

$\Rightarrow l=r\theta$

$=\frac{1}{\pi }\times \frac{{\pi }^2}{180}$
$=\frac{\pi }{180}$

$Keysteps:$ Knowing that in the relation
$l=r\theta ,\theta$ is in radian

$Method2$
We know ${360}^{\circ }$ corresponds to $2\pi r$ arc length
${360}^{\circ }\to 2\pi r$
$1^{\circ }\to \frac{2\pi r}{360}$
=$\frac{\pi r}{180}$
${\pi }^{\circ }\to \pi \times \frac{\pi r}{180}$
$=\frac{{\pi }^2}{180}\times \frac{1}{\pi }$
$=\frac{\pi }{180}$

$Keysteps:$ Logically establishing relation between arc length and degree measurement of $\theta$.