Question

use the information in the diagram to determine the radian measure of the angle φ.

Explanation:

Step1: Recall the relationship between arc - length, radius and angle in radians

The formula for the angle $\theta$ (in radians) in a circle is $\theta=\frac{s}{r}$, where $s$ is the arc - length and $r$ is the radius.

Step2: Identify the arc - length and radius from the diagram

Assume the outer - circle has a radius $r_1 = 1$ and the inner - circle has a radius $r_2=0.5$. The arc shown is a semi - circle of the inner - circle. The arc - length $s$ of a semi - circle with radius $r$ is $s=\pi r$. Here, with $r = 0.5$, $s=\frac{\pi}{2}$. The radius for the angle measurement (we can consider the path of the arc) is $r = 0.5$.

Step3: Calculate the angle in radians

Using the formula $\theta=\frac{s}{r}$, substituting $s=\frac{\pi}{2}$ and $r = 0.5=\frac{1}{2}$, we get $\theta=\frac{\frac{\pi}{2}}{\frac{1}{2}}=\pi$.

Answer:

$\pi$