Answer :

i) A regular hexagon is made up of 6 equilateral triangles.


Therefore, the green coloured triangles inside the yellow hexagon are equilateral triangles, that is all their sides are equal.


All the sides that comprises the hexagon are equal.


Hence, the inner red hexagon is also regular.


ii) Let the side of the inner red hexagon be x


the sides of the triangle will also be x


Let the side of outer yellow hexagon be y


Using Pythagoras theorem,


x2 + y2 = (2x)2


y2 = (2x)2 – x2


y2 = 4x2 – x2


y2 = 3x2


y= x√3


Hence, the outer hexagon’s side is √3 times the inner hexagon’s side.


iii) Now


Area of larger hexagon


Area of larger hexagon


And,


Area of smaller hexagon


Hence, the outer hexagon’s area is 3 times the inner hexagon’s area.


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