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 $\mathrm{\angle }AOB$ 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 $\mathrm{\angle }AOB$ is approximately 103.1 degrees (to the nearest tenth of a degree).