Question

In Exercises \(1-4,\) the angle lies at the center of a circle and subtends an arc of the circle. Find the missing angle measure, circle radius, or arc length. \(\begin{array}{ll}{\text { Angle }} & {\text { Radius }} &{\text {Arc Length }} \\ {?} & {6} & {3 \pi / 2}\end{array}\)

Short answer

The missing angle measure is \( \Theta = \pi / 4 \) radians

Step-by-step solution

Understand the problem

Given the radius \( r = 6 \) and the arc length \( s = 3\pi / 2 \), we need to find the angle \( \Theta \) subtended by the arc at the center. Here the angle is measured in radians.

Use the relationship between angle, radius and arc length

Using the formula \( \Theta = s / r \), where \( \Theta \) is the angle, \( s \) is the arc length, and \( r \) is the radius, substitute the given values for \( r \) and \( s \) into the formula to obtain \( \Theta = (3\pi / 2) / 6 \)

Calculate the value of the angle

Do the division to find the value of \( \Theta \), which equals \( \pi / 4 \) radians