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 angleAOB
is approximately blank degrees.

The solution is

Answer 1

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

\[ s = r \theta \]

where:

• \( s \) is the arc length,
• \( r \) is the radius, and
• \( \theta \) is the angle in radians.

From the problem:

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

Now, substituting the known values into the formula:

\[ 9 = 5 \theta \]

To find \( \theta \):

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

Next, to convert radians to degrees, we use the conversion factor \( \frac{180^\circ}{\pi \text{ radians}} \):

\[ \theta \text{ (degrees)} = 1.8 \times \frac{180^\circ}{\pi} \]

Calculating this:

\[ \theta \text{ (degrees)} \approx 1.8 \times 57.2958 \approx 103.1^\circ \]

Therefore, the measure of the central angle \( \angle AOB \) is approximately 103.1 degrees (to the nearest tenth of a degree).