The "sphere" is actually a half-sphere, sitting on top of a cylinder. The cylinder has height equal to the radius of the sphere.
The "octahedron" is the top half of a pyramid... if you look closely you can see the rest of the pyramid under the water. The cube is actually a cube, the water level cutting it in half and reflecting it onto itself.
There's plenty of good math ideas here:
How does the volume of the half-sphere compare to the volume of the cylinder it sits on?
Which is greater, the ratio of (volume half-sphere):(volume cylinder) or (volume cylinder):(volume whole-sphere)?
What is the ratio (volume octahedron):(volume big pyramid)?