Question

the diameter of a circle is 8 kilometers. what is the angle measure of an arc π kilometers long?
l=π km
d=8 km
give the exact answer in simplest form.

Explanation:

Step1: Find the circumference of the circle

The formula for the circumference \( C \) of a circle is \( C = \pi d \), where \( d \) is the diameter. Given \( d = 8 \) km, so \( C=\pi\times8 = 8\pi \) km.

Step2: Set up the proportion for arc length and angle

The ratio of the arc length \( l \) to the circumference \( C \) is equal to the ratio of the central angle \( \theta \) (in degrees) to \( 360^\circ \). So \( \frac{l}{C}=\frac{\theta}{360^\circ} \). We know \( l = \pi \) km and \( C = 8\pi \) km. Substituting these values, we get \( \frac{\pi}{8\pi}=\frac{\theta}{360^\circ} \).

Step3: Solve for \( \theta \)

Simplify \( \frac{\pi}{8\pi}=\frac{1}{8} \). Then \( \frac{1}{8}=\frac{\theta}{360^\circ} \). Cross - multiply: \( \theta=\frac{360^\circ}{8}=45^\circ \).

Answer:

\( 45 \)