Answer :

Case (i) for regular hexagon:


In the figure given below:



Side of regular hexagon = 2 cm


In this figure,


Radius of circle = Side of hexagon = 2cm


In triangle OAB,


OAB = OBA = 60° (OA = OB = radius = 2 cm)


If r = radius of circle


Circumference = 2 × π × r


Hence, perimeter of each circle = 2 × π × 2≈4 × 3.14 = 12.56 cm


Case (ii) for square:


The figure is shown below:



In triangle OAB,


OA = OB = radius


In triangle OAB,


OAB = OBA = 45°


Sum of all angles of a triangle = 180°


OAB + OBA + AOB = 180°


45° + 45° + AOB = 180°


90° + AOB = 180°


AOB = 180-90 = 90°


By Pythagoras theorem,


(Hypotenuse)2 = (One side)2 + (Other side)2


(AB)2 = (OA)2 + (OB)2


22 = 2 × (OA)2 (OA = OB)


(OA)2 =


OA = √2 = radius


If r = radius of circle


Circumference = 2 × π × r


Hence, perimeter of each circle = 2 × π × √2


≈4 × 3.14 = 8.87 cm


Case (iii) for rectangle drawn within the circle. The figure is displayed below:



In triangle ABC,


ABC = 90° (angle subtended in the semicircle is right angle).


By pythagoras theorem,


(Hypotenuse)2 = (One side)2 + (Other side)2


(AC)2 = (AB)2 + (AC)2


(AC)2 = 22 + 1.52


(AC)2 = 4 + 2.25 = 6.25


AC = √6.25 = 2.5 = Diameter of circle.


Radius =


If r = radius of circle


Circumference = 2 × π × r


Perimeter = 2 × π × r≈2 × 3.14 × 1.25 = 7.85 cm


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