You have a regular 6 sided polygon and a regular 3 sided polygon.

Question

If the two figures have the exact same perimeter, what is the ratio of their areas?

Answer 1

Each side of the equilateral triangle is twice the length of a side of the hexagon.

The hexagon can be divided into 6 equilateral triangles.

The triangle can be divided into 4 equilateral triangles.

Al triangles are equal in size.

The ratio of areas is hexagon : triangle :: 6 : 4

Answer 2

First find length of both polygon

Let us suppose perimeter of both polygon is 'P'.

then as we know,

Perimeter formula for 6 side (ie. hexagon) is = 6*l₁

Perimeter formula for 3 side polygon(i.e. triangle) = 3* l₂

here 'l' is the length of side of polygon.

Then from above,

Length of hexagon will be (l₁) = P/6

Length of triangle will be (l₂)= P/3

Now is the time to use area formula for both polygon.

Ratio of Hexagon to Triangle area is = (((3 sqrt (3))*l₁²)/2)/(((sqrt(3) * l₂²))/4) -----------------(1)

now plug the value of l₁ and l₂ in the equation 1 above and solve it

= solving is easy. just plugging value and calculating.

After solving it we will end up getting answer : 3/2

Answer 3

Let s = side of the hexagon, then the area of the hexagon is (3/2)s²√3.

Let S = side of the triangle; 6s = 3S, then S=2s

The area of the triangle then the area of the triangle is (1/2)s²√3

The ratio of the areas is 3:1.