The picture shows the vertices of a regular hexagon connected by lines.

i) Prove that the inner red hexagon is also regular.

ii) How much of a side of the large hexagon is a side of the small hexagon?

iii) How much of the area of the large hexagon is the area of the small hexagon?

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.