If the radius of the circle is 90 cm, and the length (L) of its arc is 25 cm, what is the degree of the measure of the arc?

So we can write

To find the degree measure of the arc, you can use the formula:

[ \text{Arc length (L)} = \frac{\text{Degree measure of arc}}{360^\circ} \times 2\pi r ]

Where:

• ( \text{Arc length (L)} ) is the length of the arc,
• ( r ) is the radius of the circle.

Rearranging the formula to solve for the degree measure of the arc:

[ \text{Degree measure of arc} = \left( \frac{L}{2\pi r} \right) \times 360^\circ ]

Now, plug in the given values:

[ \text{Degree measure of arc} = \left( \frac{25 , \text{cm}}{2 \times \pi \times 90 , \text{cm}} \right) \times 360^\circ ]

[ \text{Degree measure of arc} \approx \left( \frac{25}{2 \times 90 \times \pi} \right) \times 360^\circ ]

[ \text{Degree measure of arc} \approx \left( \frac{25}{180 \times \pi} \right) \times 360^\circ ]

[ \text{Degree measure of arc} \approx \left( \frac{25 \times 360}{180 \times \pi} \right)^\circ ]

[ \text{Degree measure of arc} \approx \left( \frac{9000}{180 \times \pi} \right)^\circ ]

[ \text{Degree measure of arc} \approx \left( \frac{9000}{180} \times \frac{1}{\pi} \right)^\circ ]

[ \text{Degree measure of arc} \approx \left( 50 \times \frac{1}{\pi} \right)^\circ ]

[ \text{Degree measure of arc} \approx \left( \frac{50}{\pi} \right)^\circ ]

[ \text{Degree measure of arc} \approx \frac{50}{\pi} \times 180^\circ ]

[ \text{Degree measure of arc} \approx \frac{50}{\pi} \times 180 ]

[ \text{Degree measure of arc} \approx \frac{50 \times 180}{\pi} ]

[ \text{Degree measure of arc} \approx \frac{9000}{\pi} ]

[ \text{Degree measure of arc} \approx 2864.788 , \text{degrees (approximately)} ]

So, the degree measure of the arc is approximately ( 2864.788^\circ ).