The radius of a circle described around a regular octagon is 2 cm, find the radius of the circle inscribed in it.

As you know, the radius of a circumscribed circle around a regular polygon can be calculated by the formula:

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

At the same time, 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:

a = R * 2sin (180 / n);

r = R * 2sin (180 / n) / 2tg (180 / n);

r = R * 2sin (180 / n) / 2 (sin (180 / n) / cos (180 / n));

r = R * cos (180 / n).

Let us find out what the radius is equal to in this case:

2 * cos (180/8) = 2 * cos 22.5 = 2 * √ ((1 + cos 45) / 2) = 2√ ((2 + √2) / 2 = √ (2 + √2).

Answer: √ (2 + √2).