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.

Answer 1

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

\[ \text{Arc Length} = r \cdot \theta \]

where:

• \( \text{Arc Length} \) is the length of the arc (AB) in centimeters,
• \( r \) is the radius of the circle in centimeters,
• \( \theta \) is the central angle in radians.

Given:

• Arc Length \( = 9 \) cm
• Radius \( r = 5 \) cm

We can rearrange the formula to solve for \( \theta \):

\[ \theta = \frac{\text{Arc Length}}{r} \]

Plugging in the values:

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

Now, to convert radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):

\[ \theta_{\text{degrees}} = \theta \times \frac{180}{\pi} \]

Calculating this:

\[ \theta_{\text{degrees}} = 1.8 \times \frac{180}{\pi} \approx 1.8 \times 57.2958 \approx 103.1 \]

Therefore, the measure of central angle \( AOB \) is approximately 103.1 degrees.