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}

2^{2} = 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} = 2^{2} + 1.5^{2}

∴ (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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