Using JSON notation, how do I create a public method in javascript?

As T.J. Crowder says... that's not JSON. I think maybe the problem is that the getArea property of your circle object is being set to whatever the getArea function returns. Which means that circle.getArea() won't work because circle.getArea is a number. This is wrong but I think its a step closer to what you are trying to achieve (I'll show a better way after this):

var circle = {
  radius : 9,
  getArea : function(){
    return Math.PI * 9 * 9;  
  }
};

alert(circle.radius);
alert(circle.getArea());

The important thing above is to note the way I define the getArea function inside the object. This can be useful, but the problem is that the get area function doesn't have access to the radius variable, because objects like circle don't have their own scope. If you try :

var myObj = {test:this};

alert(myObj.this);

You'll see [object DOMWindow]

So how do you create an object Circle with a radius property and a getArea method? There are a many ways. One way is like this:

function Circle(r){
   var c = this;
   this.radius = r;
   this.getArea = function(){
     return Math.PI * c.radius * c.radius;
   }
}

var circle = new Circle(10);
document.write(circle.getArea()+"<br/>");

circle.radius = 20;

document.write(circle.getArea()+"<br/>");

var otherCircle = new Circle(1);
document.write(otherCircle.getArea()+"<br/>");

I've created a jsFiddle of the above.

This can also be achieved using prototype property:

function Circle(r){
   this.radius = r;
}

Circle.prototype.getArea = function(){
   return Math.PI * this.radius * this.radius;
}

var circle = new Circle(10);
document.write(circle.getArea()+"<br/>");

circle.radius = 20;

document.write(circle.getArea()+"<br/>");

var otherCircle = new Circle(1);
document.write(otherCircle.getArea()+"<br/>");

There are no such thing as private method or property in javascript.

Everyting is public but common practise is to treat anything beginning with _ as private, but thats only convention, not enforced byt the language.

To have private methods and properties you need real classes to define them and javascript does not have real classes.