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. \(\begin{array}{ll}{\text { Angle }} & {\text { Radius }} &{\text {Arc Length }} \\ {?} & {14} & {7 }\end{array}\)

Short answer

The unknown angle at the center of the circle is 0.5 radians or approximately 28.65 degrees.

Step-by-step solution

Understand what's given and what's needed

The problem provides us with the radius of the circle (14 units), and the length of the arc (7 units), so the task is to find out what the angle at the center (in radians) is. You might prefer calculating the answer in degrees instead of radians, but you can easily convert between the two units at the end.

Cover the formula that relates all three

The relationship between the arc length (L), radius (r), and the angle (θ) is expressed as \(L = r \cdot \theta\). Since we need to find the angle, we rearrange the formula to become \( \theta = \frac{L}{r} \).

Replace provided values in the equation

Substitute the provided radius (r = 14 units) and arc length (L = 7 units) into the equation. The equation becomes \( \theta = \frac{7}{14} = 0.5 \) radians.

Convert the answer into degrees if necessary

If the angle is desired in degrees instead of radians, then you would need to convert it. The conversion is done by multiplication with \(\frac{180}{\pi}\). Therefore, the angle in degrees would be \(0.5 \times \frac{180}{\pi} \approx 28.65\) degrees.