The polygon has area 7/2 * s^2 * cot( pi /14), where s is the side length.

The rectangle has height s * cot( pi /14), so area s^2 * cot(pi/14)

We can get both these from the formula for the length of the apothem which is s / (2 * tan( pi / N ) ) for an N-sided polygon. This is half the height of the rectangle, and it is the height of each of the 14 triangles with base s that make up the polygon.

So, the ratio is 7/2 : 1, that is, 7 : 2.