Calculate the volume of the spherical shell if the radii of the top and bottom circles are given as $r_{1}$ and $r_{2}$, and the height of the spherical shell is $h$.

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Note: The term "gömbréteg" in Hungarian is translated as "spherical shell" in English. If you meant a spherical cap (a portion of a sphere cut off by a plane), please clarify, as the formula for the volume of a spherical cap is different.
The volume of a spherical layer can be obtained as the difference between the volumes of two spherical segments. Let the radii of the base circles of the first and second spherical segments be $r_{2}$ and $r_{1}$, and their heights be denoted by $h_{2}$ and $h_{1}$, respectively. Let the radius of the sphere be $r$.

The sought volume is

$$
t=t_{2}-t_{1}
$$

where

$$
\begin{aligned}
& t_{2}=\frac{\pi h_{2}^{2}}{3}\left(3 r-h_{2}\right) \\
& t_{1}=\frac{\pi h_{1}^{2}}{3}\left(3 r-h_{1}\right)
\end{aligned}
$$

From this,

$$
t=\pi r\left(h_{2}^{2}-h_{1}^{2}\right)-\frac{\pi}{3}\left(h_{2}^{3}-h_{1}^{3}\right) .
$$

The quantities $r, h_{2}$, and $h_{1}$ can be calculated using the following system of equations:

$$
\begin{gathered}
h_{2}-h_{1}=h \\
h_{2}\left(2 r-h_{2}\right)=r_{2}^{2} \\
h_{1}\left(2 r-h_{1}\right)=r_{1}^{2}
\end{gathered}
$$

The simplification we intend to present here facilitates the regular procedure by making the solution of the system of equations (1) for $r, h_{2}$, and $h_{1}$ unnecessary, and further, the substitution of these values into the formula for $t$ can be omitted.

Indeed,

$$
\begin{gathered}
t=\frac{\pi h}{3}\left\{3 r\left(h_{2}+h_{1}\right)-\left(h_{2}^{2}+h_{2} h_{1}+h_{1}^{2}\right\}=\right. \\
=\frac{\pi h}{3}\left\{2 r\left(h_{2}+h_{1}\right)-\left(h_{2}^{2}+h_{1}^{2}\right)+r\left(h_{2}+h_{1}\right)-\left(h_{2} h_{1}\right)\right\}
\end{gathered}
$$

But from the last two equations of (1),

$$
2 r\left(h_{2}+h_{1}\right)-\left(h_{2}^{2}+h_{1}^{2}\right)=r_{2}^{2}+r_{1}^{2}
$$

If we add the square of the first equation of (1) to this, we get

$$
2 r\left(h_{2}+h_{1}\right)-2 h_{2} h_{1}=r_{2}^{2}+r_{1}^{2}+h^{2}
$$

$$
r\left(h_{2}+h_{1}\right)-h_{2} h_{1}=\frac{1}{2}\left(r_{2}^{2}+r_{1}^{2}+h^{2}\right)
$$

After substituting the values from equations (2) and (3),

$$
t=\frac{\pi h}{3}\left\{r_{2}^{2}+r_{1}^{2}+\frac{1}{2}\left(r_{2}^{2}+r_{1}^{2}+h^{2}\right)\right\}
$$

and finally, $\left.t=\frac{\pi h}{6}\left\{h^{2}+3 r_{1}^{2}+3 r_{2}^{2}\right)\right\}$, the familiar formula.