Γ is a circle with radius 10 and center O. A and B are points on Γ such that the arc length ABˆ=13. What is the area of sector AOB?
s = rθ
a = 1/2 r^2 θ = 1/2 r^2 (s/r)
you know r and s, so plug and chug
To find the area of sector AOB, we need to know the angle subtended by the arc AB at the center O.
The formula to find the area of a sector is:
Area = (θ/360) * π * r^2
Where θ is the central angle in degrees, π is a constant (approximately equal to 3.14159), and r is the radius of the circle.
To find θ, we can use the formula:
θ = (Arc Length / Circumference) * 360
Here, the arc length is given as 13, and the circumference of a circle can be found using the formula:
Circumference = 2 * π * r
Substituting the values into the formulas, we can solve for θ and then calculate the area of sector AOB.
Let's calculate step by step:
1. Calculate the circumference of the circle:
Circumference = 2 * π * 10 = 20π
2. Calculate the central angle θ:
θ = (Arc Length / Circumference) * 360
= (13 / (20π)) * 360
= (13 / 20) * 360π
≈ 226.195 radians (or approximately 226°) [rounded to 3 decimal places]
3. Calculate the area of sector AOB:
Area = (θ/360) * π * r^2
= (226.195 / 360) * π * 10^2
≈ 178.097 square units [rounded to 3 decimal places]
Therefore, the area of sector AOB is approximately 178.097 square units.