Circle approximation

The underlying JavaScript code

var brd = JXG.JSXGraph.initBoard('box', {originX: 300, originY: 300, grid:true, unitX: 50, unitY: 50});
var n = brd.createElement('slider',[[-5.0,5],[4.5,5],[3,4,96]],{name:'n',snapWidth:1});

var p0 = brd.createElement('point',[0,0]);
var p1 = brd.createElement('point',[4,0]);
var circ = brd.createElement('circle',[p0,p1]);
var rot = brd.createElement('transform', [function() {return Math.PI*2.0/n.Value();},p0], {type:'rotate'});  // angle, rotation center
var ptmp = brd.createElement('point',[0,0],{visible:false,withLabel:false});  // dummy point for the rotation
var cOut = brd.createElement('curve',[[],[]],{fillColor:'#0033ff',highlightFillColor:'#0033ff',fillOpacity:0.5,highlightFillOpacity:0.5});
var cIn = brd.createElement('curve',[[],[]],{fillColor:'#ff3300',highlightFillColor:'#ff3300',fillOpacity:0.5,highlightFillOpacity:0.5});

var tCirc = brd.createElement('text',[-5,-4.0,function(){
    return 'Area of the circle = ' + (circ.getRadius()*circ.getRadius()*Math.PI).toFixed(4);
  }],{fontSize:'20px'});
var tIn = brd.createElement('text',[-5,-4.5,function(){
    return 'Area of the inscribed polygon = ' + (n.Value()*circ.getRadius()*circ.getRadius()*Math.sin(Math.PI/n.Value())).toFixed(4);
  }],{fontSize:'20px'});
var tOut = brd.createElement('text',[-5,-5,function(){
    return 'Area of the circumscribed polygon = ' + (n.Value()*circ.getRadius()*circ.getRadius()*Math.tan(Math.PI/n.Value())).toFixed(4);
  }],{fontSize:'20px'});

cIn.updateDataArray = function() {
  var i;
  this.dataX = [p0.X()+circ.getRadius()];
  this.dataY = [p0.Y()];
  ptmp.setPositionDirectly(JXG.COORDS_BY_USER,p0.X()+circ.getRadius(),p0.Y());
  for (i=0;i<n.Value();i++) {
    rot.applyOnce(ptmp);
    this.dataX.push(ptmp.X());
    this.dataY.push(ptmp.Y());
  }
}

cOut.updateDataArray = function() {
  var i;
  var s = circ.getRadius()/Math.cos(Math.PI/n.Value());
  this.dataX = [p0.X()+s];
  this.dataY = [p0.Y()];
  ptmp.setPositionDirectly(JXG.COORDS_BY_USER, p0.X()+s,p0.Y());
  for (i=0;i<n.Value();i++) {
    rot.applyOnce(ptmp);
    this.dataX.push(ptmp.X());
    this.dataY.push(ptmp.Y());
  }
}

brd.update();