A triangle has sides $a=20 \mathrm{~cm}, b=25 \mathrm{~cm}, c=30 \mathrm{~cm}$. What is the length of one side of a regular octagon with the same area as the triangle?
Let $t$ be the area of the triangle. Then

$$
t=\sqrt{s(s-a)(s-b)(s-c)}=248.039 \mathrm{~cm}^{2}
$$

If the side of the regular octagon we are looking for is $x$, then

$$
t=8 \cdot \frac{x^{2}}{4} \cdot \operatorname{ctg} \frac{180^{\circ}}{8}
$$

from which

$$
c=\sqrt{\frac{t \operatorname{tg} 22^{\circ} 30^{\prime}}{2}}=7.167 \mathrm{~cm}
$$

(József Kussinszky, Pécs.)