## Task 4 - 070924

A sphere is circumscribed around a regular octahedron.

Calculate the ratio of the surface areas of both figures!
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 $\pi$.