Unizor - Geometry3D - Problems on Similarity

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Problems on
Similarity in 3-D Space

1. Prove that all cubes are similar.

2. Prove that all regular tetrahedrons are similar.

3. Prove that all spheres are similar.

4. Consider a sphere inscribed into a cube of a unit size. It takes up certain volume in the cube, while certain part of a cube's volume remains free.
Now let's take this sphere out from a cube, take certain number of spheres with a diameter equaled to 1/Nth of the original sphere and fill the cube with these smaller spheres positioning these spheres in rows, columns and layers, one strictly above the other. They will also take up some space, while certain part of a cube's volume will remain free.
In which of these cases the free part of a cube will be larger by volume?

5. Consider two persons drinking martini from the same conical glasses. One takes the glass filled with martini to the brim, while another said to a barman that she wants only half a glass. Barman fills her glass to half its height. How much martini is in this glass relatively to a full glass of her partner?

6. Specify requirements for two right rectangular prisms that assure their similarity.