The perimeter of a regular hexagon with vertices on a circle is 24 centimetres.

i) What is the perimeter of a square with vertices on this circle?

ii) What is the perimeter of a square with vertices on a circle of double the diameter?

iii) What is the perimeter of an equilateral triangle with vertices on a circle of half the diameter of the first circle?

Given:

Perimeter of Hexagon = 24 cm

Let a = side of Hexagon

But, Perimeter of Hexagon = 6 × side of Hexagon

24 = 6 × a

∴ a = 24/6=4

As can be seen from figure,

Radius of circle = Side of Hexagon

∴ Radius of circle = a = 4 cm

i) Diameter of circle = 2 × radius

= 2 × 4

= 8 cm

As seen in figure,

Diameter of circle = Diagonal of square

Diagonal of square = 8 cm

For a square,

If side = a cm

Diagonal of square = = 8

∴

Perimeter of square = 4 × side of square

=

=

= cm

ii) Diameter of circle =2 × Initial diameter

=2 × 8=16 cm

As seen in figure,

Diameter of circle = Diagonal of square

Diagonal of square = 16 cm

For a square,

If side = a cm

Diagonal of square = √2a = 16

∴

Perimeter of square = 4 × side of square

=

=

= cm

iii)

Diameter of circle =1/2 × Initial diameter

=

∴ Radius of circle = cm = 2 cm

As proved earlier, the circumcentre of an equilateral triangle is the same as its centroid.

Also, centroid divides equilateral in the ratio 2:1

∴ radius =

∴ 2 =

∴ height of triangle = × 2 = 3

But, for an equilateral triangle,

Height of triangle =

3 =

∴ side of triangle = 3

Perimeter of equilateral triangle =

=

=

= cm