## Task 3 - 090913

In the figure, a convex body bounded by plane surfaces is shown in top, front, and side views.

A body $K$ bounded by plane surfaces is called convex if for each of its boundary surfaces $F$ the following holds: If $\varepsilon$ is the plane in which $F$ lies, then $K$ is entirely in one of the two half-spaces into which the space is divided by $\varepsilon$.

The outlines of the depicted body are squares with side length $a$ in the top, front, and side views.

Construct or describe such a body, and calculate its volume!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0929.jpg?height=708&width=886&top_left_y=737&top_left_x=585)
To find the solution, it is useful to imagine the body inscribed in a cube. Furthermore, three points lying on a straight line in the projection lie in a plane in the original (and as one can easily see from the distances, even on corners). The face diagonals, after successfully determining some side assignments, quickly make the problem unique.

The volume of the non-regular octahedron is that of two quadrilateral pyramids over the area $a \cdot \sqrt{2} \cdot a$ with the heights of half the face diagonals, thus $2 \cdot \frac{1}{3} \cdot \frac{a}{\sqrt{2}} \cdot a \cdot \sqrt{2} a=\frac{2}{3} \cdot a^{3}$.