Given a point, a "radius" and an arc, how do I find the resulting point?

When you already know the center point for the rotation as well as the angle, this is a simple and straightforward transformation to apply.

Fundamentally, the calculations are based on a 2-Dimensional, 2x2 rotation matrix.

 // Where "T" is the rotation value theta, or θ
 ┌                 ┐
 |cos(T)    -sin(T)|
 |sin(T)    cos(T) |
 └                 ┘

When you apply this to a given value, it will look something like this:

 // C#
 public static Vector2 RotatePoint(Vector2 inputPoint, Vector2 center, float degrees)
 {
     // First, convert degrees to radians
     float radians = degrees * Mathf.Deg2Rad;
     
     // Then, create the output
     Vector2 outputPoint = inputPoint;
     
     // Next, shift the input based on the center of rotation
     Vector2 shiftedPoint = inputPoint - center;
     
     // Finally, apply the values from the matrix to the X and Y
     outputPoint.x = center.x + (shiftedPoint.x * Mathf.Cos(radians) - shiftedPoint.y * Mathf.Sin(radians));
     outputPoint.y = center.y + (shiftedPoint.y * Mathf.Cos(radians) + shiftedPoint.x * Mathf.Sin(radians));
     
     return outputPoint;
 }