Find the measure of the angle

Given a circle with center O, and two points A and B on the circumference of the circle, form the angle AOB. If the radius of the circle is 5 cm and the arc AB measures 8 cm, find the measure of the angle AOB in radians and degrees.

Step 1: Find the circumference of the circle using the formula:

$ C = 2 \pi r $

Step 2: Substitute the radius into the formula:

$ C = 2 \pi (5) = 10 \pi \text{ cm} $

Step 3: Find the fraction of the circumference that the arc AB represents:

$ \frac{8}{10\pi} = \frac{4}{5 \pi} $

Step 4: Multiply this fraction by the total measure of the circle in radians (2π):

$ \text{Angle in radians} = 2 \pi \times \frac{4}{5 \pi} = \frac{8}{5} \text{ radians} $

Step 5: To convert radians to degrees, use the fact that π radians = 180 degrees:

$ \text{Angle in degrees} = \frac{8}{5} \times \frac{180}{\pi} \approx 91.56° $

Consider the same circle with radius 5 cm, and arc length AB as 8 cm. We need to find the angle AOB in radians and degrees.

1. Calculate the circumference:

$ C = 2 pi r = 10 pi ext{ cm} $

2. The fraction of the circle’s circumference that arc AB represents:

$ frac{8}{10pi} approx 0.2546 $

3. Multiply this by 2π to find the angle in radians:

$ ext{Angle in radians} = 0.2546 imes 2pi approx 1.6 ext{ radians} $

4. Convert to degrees:

$ ext{Angle in degrees} = 1.6 imes frac{180}{pi} approx 91.66° $

For a circle with radius 5 cm and arc length 8 cm,

1. Circumference:

$ C = 2 pi r = 10 pi $

2. Arc length fraction:

$ frac{8}{10pi} = frac{4}{5 pi} $

3. Angle in radians:

$ frac{8}{5} $

4. Angle in degrees:

$ approx 91.56° $

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