In a circle with a radius of 5cm and is centered at point O, angle AOB intercepts arc AB. Arc AB has a length of 10cm. What is the degree measure of angle AOB? Round to the nearest hundredth.
arc length of a circle = rØ, where Ø is in radians
5Ø = 10
Ø = 2 radians or appr 114.59°
To find the degree measure of angle AOB, we can use the arc length formula:
Arc Length = (Angle / 360) * (2 * π * r)
where r is the radius of the circle.
In this case, the arc length AB is 10cm, and the radius is 5cm.
Plug in the values into the formula:
10 = (Angle / 360) * (2 * π * 5)
Now, let's solve for the angle:
Multiply both sides of the equation by (360 / (2 * π * 5)):
10 * (360 / (2 * π * 5)) = Angle
Simplify the expression on the left:
(10 * 360) / (2 * π * 5) = Angle
Multiply:
3600 / (10 * π) = Angle
Divide:
360 / π = Angle
Now, let's substitute the value of π (pi) to get the decimal approximation:
360 / 3.14159 = Angle
Angle ≈ 114.5916 (rounded to the nearest hundredth)
Therefore, the degree measure of angle AOB is approximately 114.59°.