Question

How do you find the radian measure of the central angle of a circle of radius 80 kilometers that intercepts an arc length 160 kilometers?

Answer 1

Hint: This problem deals with finding the radian measure. One radian is the measure of a central angle subtended by an arc that is equal in length to the radius of the circle. When no symbol is used, radians are assumed. When degrees are the unit of angular measure, the symbol $^ \circ $ is written. Note that the radian is a derived unit in the international system of units.

Complete step-by-step solution:
Radian describes the plane angle subtended by a circular arc, as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.
So here given that the radius of the circle $r$ is 80 kilometers.
The arc length of the given circle is 160 kilometers.
Take the arc length as $\theta = 160$
The central angle in radians is given by $2\pi $.
The angles to arc length is given by:
$ \Rightarrow \dfrac{\theta }{r}$
Where the circumference of the circle is given by: $2\pi r$
So the radian measure is given by:
$ \Rightarrow \dfrac{\theta }{r} = \dfrac{{160}}{{80}}$
$ \Rightarrow \dfrac{\theta }{r} = 2$

The radian measure is given by $\dfrac{\theta }{r}$ which is equal to 2.

Note: Please note that to convert from degrees to radian, multiply the angle by $180$. To convert from radian to degrees, multiply the angle with $180$. In terms of degrees, one radian is approximately ${57.3^ \circ }$. Likewise, an angle of ${1^ \circ }$ is approximately 0.017 radians.