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. \(\begin{array}{ll}{\text { Angle }} & {\text { Radius }} &{\text {Arc Length }} \\ {175^{\circ}} & {?} & {10 }\end{array}\)

Short answer

The radius of the circle is approximately 3.26 units.

Step-by-step solution

Defining Known Value

From the problem, it is known that the measure of the angle is \(175^\circ\) and the length of the arc subtended by this angle is \(10\) units.

Convert Degree to Radians

Since the formula that relates angle, arc length and radius involves angle in radians, we need to convert the angle from degrees to radians. We know that \(1^\circ = \frac{\pi}{180}\) radians. Hence, \(175^\circ = 175 \times \frac{\pi}{180} = \frac{35\pi}{36}\) radians.

Implement the Relation

We have a formula that connects the three parameters of the equation: \[Arc\_Length = Radius \times Angle\] Hence, we can manipulate the formula to calculate the radius: \[Radius = \frac{Arc\_Length}{Angle}\]

Substitute the Known Values

Substitute the known values into the formula to find the radius: \[Radius = \frac{10}{\frac{35\pi}{36}} = \frac{360}{35\pi} \approx 3.26\] units