The side of a regular polygon is a = 3cm, and the radius of the inscribed circle is r = 2cm.

The side of a regular polygon is a = 3cm, and the radius of the inscribed circle is r = 2cm. Find the radius of the circumscribed circle.

As you know, the radius of a circle that is inscribed in a regular polygon can be calculated by the formula:

r = a / 2tg (180 / n).

Then we get that:

2tg (180 / n) = a / r;

tg (180 / n) = a / 2r.

Or, substituting the given from the condition of the problem:

tg (180 / n) = 3 / (2 * 2) = 3/4.

In order to calculate the radius of the circumscribed circle, you should use the formula:

R = a / 2sin (180 / n).

Find the value of sin (180 / n), knowing the tangent:

1 + ctg2 a = 1 / sin2 a;

1 + (1 / tan a) ^ 2 = 1 / sin ^ 2 a;

1 + (4/3) ^ 2 = 1 / sin ^ 2 a;

sin a = √ (1: (25/9));

sin a = 0.6;

R = 3 / (2 * 0.6) = 2.5.

Answer: 2.5 cm.