To determine the ratio of the surface areas of the two figures, these areas must be known. For the surface area of a sphere, the following applies:

$$
A_{\text {Sphere }}=4 \pi r^{2}
$$

The formula for calculating the surface area of a regular octahedron is:

$$
A_{\text{Octa}}=2 a^{2} \sqrt{3}
$$

Now we need to establish the relationship between the radius $r$ of the sphere and the edge length $a$ of the octahedron. If we bisect the octahedron exactly between the two vertices, we obtain a square cross-section with side length $a$, which is also the edge length of the given octahedron.

The radius $r$ of the circumscribed sphere is exactly half the length of the diagonal of this cross-sectional square, i.e., $r=\frac{a \sqrt{2}}{2}$.

To determine the surface area of the sphere, we substitute the radius $r$ and thus obtain for the surface area of the sphere:

$$
A_{\text {Sphere }}=2 \pi a^{2}
$$

The desired ratio is

$$
\frac{A_{\text {Octa }}}{A_{\text {Sphere }}}=\sqrt{3}: \pi
$$

The surface area of the regular octahedron is to the surface area of the circumscribed sphere as $\sqrt{3}$ to $\pi$.
## Task 4 - 070924

A sphere is circumscribed around a regular octahedron.

Calculate the ratio of the surface areas of both figures!