Circular Sector: Finding the Radius and Angle
Here is the problem:
We are given the area of the circular segment, A, and the arc length of the segment, s. What is the radius, r, and the angle, θ?
The arc length is calculated as: s = θ * r
The area is calculated as: A = ½ * θ * r^2
We have the system of equations:
A = ½ * θ * r^2
s = θ * r
Divide A by s:
A / s= (½ * θ * r^2) / (θ * r)
A / s = r / 2
2 * A / s = r
Then
s = r * θ
θ = s / r = s^2 / (2 * A)
In summary:
r = 2 * A / s
θ = s / r = s^2 / (2 * A)
Note that the angle is in radians.
Example
Example 1:
s = 4, A = 30
r = (2 * 30) / 4 = 15
θ = 4 / 15 ≈ 0.266666667
Example 2:
s = 10.5, A = 31.8
r = (2 * 30) / 4 = 212/35 ≈ 6.057142857
θ = 10.5 / (212/35) = 735/424 ≈ 1.733490566
Eddie
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