Given a circle with unit radius a Construct a hexagon inscri

Given a circle with unit radius:

(a) Construct a hexagon inscribed within the circle.

(b) Construct a hexagon circumscribed about the circle.

(c) Use the two hexagons to produce lower and upper bounds for .

Solution

(a) we have a unit radius circle.

First draw a diameter horizontally. Now locate it\'s end points as two vertices of hexagon.

Now take one end point and make small semicircle at there, and cut arc of 60 on it. Extend the line so that it cut the circle. This would be another vertex.

Now set your compass to the length of this cut and mark another cut on this circle. This would be another vertex.

Same as repeat for other half semicircle.

You will get 6 vertices.

(b) wet have given a circle of radius of 1.

Now draw another circle with concentric it of radius 2/3.

Now take one point on outer circle and draw tangents to inner circle from that point.

Now extend these tangents so that it cut outer circle at other two points. Now you have three vertices of hexagon.

Same repeat by taking another point on outer circle just opposite to it. (180°)

You will get other three points as well.

And with six vertices, join them. You get hexagon.