Question

What is the angle in both radians and degrees determined by an arc of length 2pi meters on a circle of radius 8 meters?

Answer 1

$\bf \textit{arc's length}\\\\ s=r\theta ~~ \begin{cases} r=radius\\ \theta = angle~in\\ \qquad radians\\ ------\\ r=18\\ s=2\pi \end{cases}\implies 2\pi =18\theta \implies \cfrac{2\pi }{18}=\theta\implies \cfrac{\pi }{9}=\theta \\\\ -------------------------------$

$\bf \textit{arc's length}\\\\ s=\cfrac{\theta \pi r}{180} ~~ \begin{cases} r=radius\\ \theta = angle~in\\ \qquad degrees\\ ------\\ r=18\\ s=2\pi \end{cases}\implies 2\pi =\cfrac{\theta \pi 18}{180} \\\\\\ 2\pi =\cfrac{\theta \pi }{10}\implies \cfrac{2\pi (10)}{\pi }=\theta \implies 20=\theta$

Answer 2

The angle in both radians and degrees determined by an arc of length 2pi meters on a circle of radius 8 meters is π/4 rad and 45degrees respectively.

The length of an arc is expressed using the formula below as shown:

$L =\frac{\theta}{360} \times 2 \pi r$

r is the radius of the arc = 8 m

theta is the angle subtended by the arc

L is the length of the arc = 2π meters

Substitute the given parameters into the formula as shown:

$2 \pi =\frac{\theta}{360} \times 2 \pi (8)\\2 = \frac{\theta}{360} \times 16\\16 \theta = 720\\\theta = \frac{720}{16}\\\theta = 45^0$

Convert 45 degrees to radians

Using the conversion rate 180 degrees = π rad

180 degrees = π rad

45 degrees = x

x= 45π/180

x = π/4 rad

Hence the angle in both radians and degrees determined by an arc of length 2pi meters on a circle of radius 8 meters is π/4 rad and 45degrees respectively.

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