What is the measure of central angle AOB to the nearest tenth of a degree?

Question

An image shows a circle with central angle A O B. O is at the center of the circle. Points A and B are on the circumference of the circle making A O and B O radii of the circle. A B is an arc along the circumference of the circle and is labeled showing the arc length of A B is 9 centimeters. A O is labeled 5 centimeters.

The measure of <AOB
is approximately blank degrees.

The solution is

Answer 1

To find the measure of the central angle \(AOB\) in degrees, we can use the formula that relates the arc length \(s\), the radius \(r\), and the central angle \(\theta\) in radians:

\[ s = r \theta \]

Given:

• Arc length \(s = 9 \text{ cm}\)
• Radius \(r = 5 \text{ cm}\)

First, we can substitute the given values into the formula to find \(\theta\):

\[ 9 = 5 \theta \]

Now, solve for \(\theta\):

\[ \theta = \frac{9}{5} = 1.8 \text{ radians} \]

Next, we'll convert \(\theta\) from radians to degrees. We use the conversion factor \(180^\circ/\pi\) to do this:

\[ \text{Degrees} = \theta \cdot \frac{180^\circ}{\pi} \approx 1.8 \cdot \frac{180^\circ}{\pi} \]

Calculating this gives:

\[ \text{Degrees} \approx 1.8 \cdot \frac{180}{3.14159} \approx 1.8 \cdot 57.2958 \approx 102.8^\circ \]

Rounding to the nearest tenth of a degree, the measure of central angle \(AOB\) is approximately:

\[ \boxed{102.8} \text{ degrees} \]