# In the picture be

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,

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 = 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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