About regular heptagons, the side lengths of which are proportional to the numbers 5 and 4

About regular heptagons, the side lengths of which are proportional to the numbers 5 and 4, are described in a circle. The radius of one of them is 3 cm larger than the radius of the other. Calculate the lengths of the diameters of the given circles

The radius of a circle circumscribed about a regular N-gon can be calculated by the formula

ρ = a / (2 * sin (180 ° / N),

where a is the length of one side of the N-gon.

Let the lengths of the sides of the given heptagons be 5x and 4x, respectively. Then the radii of the circles described around them are equal, respectively

R = 5x / (2 * sin (180 ° / 7);

r = 4x / (2 * sin (180 ° / 7),

and the ratio of these radii is

R / r = 5/4,

or

R = 1.25 * r.

Since by condition R = r + 3 cm, then

r + 3 = 1.25 * r,

where

0.25 * r = 3;

r = 12 cm – the radius of the small circle;

R = 1.25 * r = 1.25 * 12 = 15 cm – the radius of the great circle.

Then the diameters of these circles are equal respectively

d = 2 * r = 2 * 12 = 24 cm;

D = 2 * R = 2 * 15 = 30 cm.