Sunday, May 5, 2024

Circular Sector: Finding the Radius and Angle

 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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