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CAT Relating Arc Lengths to Radius - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

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4 Questions around this concept.

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The length of an arc of a circle of radius 5 cm subtending central angle measuring 45° is:

$\frac{\pi}{4}\ cm$

$\frac{5\pi}{2}\ cm$

$\frac{5\pi}{4}\ cm$

None of these

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Concepts Covered - 1

Relating Arc Lengths to Radius

An arc length s, is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius.

This ratio, called the radian measure, is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle (θ) as the ratio of the arc length s to the radius r.

$\begin{aligned} \boldsymbol{s}&=r \boldsymbol{\theta} \\ \boldsymbol{\theta}&=\frac{\boldsymbol{s}}{\boldsymbol{r}} \end{aligned}$

$\text{If } \boldsymbol{s} = \boldsymbol{r} \text{ then } \boldsymbol{\theta}=\frac{ \boldsymbol{r}}{ \boldsymbol{r}}=1\; radian.$

(a) In an angle of 1 radian, the arc length s equals the radius r.

(b) An angle of 2 radians has an arc length s = 2r.

To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is 2πr (r is thae radius). The smaller circle then has circumference 2π(2) = 4π and the larger has circumference 2π(3) = 6π. Now, draw a 45º angle on the two circles,

Notice what happens if we find the ratio of the arc length divided by the radius of the circle.

$\begin{aligned} \text { Smaller circle: } \frac{2\times\frac{\pi}{4} }{2}=\frac{1}{4} \pi \\ \text { Larger circle: } \frac{3\times\frac{\pi}{4} }{3}=\frac{1}{4} \pi \end{aligned}$