In the picture below, a regular hexagon, square and a rectangle are drawn with their vertices on a circle. Calculate the perimeter of each circle.
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