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If you want a simple solution just randomize the radius too:

private Point CalculatePoint()
{
    var angle = _random.NextDouble() * Math.PI * 2;
    var radius = _random.NextDouble() * _radius;
    var x = _originX + radius * Math.Cos(angle);
    var y = _originY + radius * Math.Sin(angle);
    return new Point((int)x,(int)y);
}

That however results in your points being more concentrated towards the center of the circle:

In order to get an uniform distribution make the following change to the algorithm:

var radius = Math.Sqrt(_random.NextDouble()) * _radius;

Which will give the following result:

For more information check the following link: MathWorld - Disk Point Picking.

And finally here's a simple JsFiddle demonstration comparing both version of the algorithm.