## 117. Math Puzzle $2 / 75$

Above a window (width $120 \mathrm{~cm}$) there is a circular arch that starts at both sides with the top edge of the window and creates a height of $12 \mathrm{~cm}$ at the center.

Calculate the radius of the arch.
Let the upper window edge be $f=120 \mathrm{~cm}$, and the height difference $h=12 \mathrm{~cm}$. From the shaded triangle, the radius follows as

$$
R^{2}=(R-12)^{2}+\left(\frac{120}{2}\right)^{2} \quad, \quad 0=-24 R+144+3600
$$

$R=156 \mathrm{~cm}$. The vault radius is $156 \mathrm{~cm}$.