Relationship of n-sided and 2n-sided polygons sharing the same circumscribed circle

						var board = JXG.JSXGraph.initBoard('box1', {
boundingbox: [-10, 10, 10, -10],showNavigation:false, showcopyright: false, axis: false,
});
var p1 = board.create('point', [-1, 0.0],{name:'O',size:1,label:{offset:[-20,0]}});
var p2 = board.create('point', [6.5, 0.0],{name:'',face:'',});

var c1 = board.create('circle', [p1, p2]);
var n = 12, s=[];
var a1 = 2*Math.PI/n;
for(var i=1;i<=n;i++){
var t1 = board.create('transform', [a1*i, p1], {type: 'rotate'});

var p3 = board.create('point', [p2, t1], {name:'',face:'',color: 'blue'});
s.push(p3);
}
for(var i=0;i

The image appears in the following posts

  1. Archimedes' approximation of Pi is by calculating the perimeter of regular polygons to determine the range of Pi. The more the number of sides, the more the perimeter of the polygon close to the circumference of its inscribed or circumscribed circle.

    Through doubling the sides of a hexagon to a 96-gon regular polygon, Archimedes found the ratio of the circumference of any circle to its diameter is less than 22/7 but greater than 223/71, i,.e.,

    \dfrac{223}{71} < \pi < \dfrac{22}{7}

    The lower limit of the approximation \dfrac{223}{71} is determined by the ratio of the perimeter of an inscribed 96-gon regular polygon to the diameter of its circumscribed circle. The upper limit \dfrac{22}{7} is determined by the ratio of a circumscribed 96-gon regular polygon to its inscribed circle. The method lies in 3 theorems. First, the ratio of the circumference of any circle to its diameter is constant. Second, the perimeter of an inscribed polygon is less than the circumference of its circumscribed circle. Third, the perimeter of a circumscribed polygon is greater than the circumference of its inscribed circle. So to prove the above approximation, implies proof of the following inequality,

    \dfrac{223}{71} < \dfrac{P_{96}}{R} <\pi < \dfrac{p_{96}}{r} < \dfrac{22}{7}

    Here we are going to skip the three preconditions, prove the above approximation directly.

Triangle posted 1 year ago

Scroll to Top