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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