What objects can be placed in a cube container with a length of 1 meter?

Question

What objects can be placed in a cube container with a length of 1 meter? The thickness of the container wall is negligible

1. A sphere with a diameter of 0.99 meters

2. Tetrahedra with all edges of 1.4 meters in length

3. A cylinder with a bottom diameter of 0.01 meters and a height of 1.8 meters

4. A cylinder with a bottom diameter of 1.2 meters and a height of 0.01 meters

I drew a cube with an edge length of 1 meter：

Clear["Global`*"];
c = {0, 0, 0};
d = {1, 0, 0};
b = {0, 1, 0};
a = b + d;
h = 1;
c1 = {0, 0, h};
d1 = d + c1
a1 = a + c1
b1 = b + c1
labels = {Text[Style[A, 12, FontFamily -> "Times"], a, {-1, -1}], 
   Text[Style[B, 12, FontFamily -> "Times"], b, {1, 1}], 
   Text[Style[C, 12, FontFamily -> "Times"], c, {1, 1}], 
   Text[Style[D, 12, FontFamily -> "Times"], d, {-2, 0}], 
   Text[Style[A1, 12, FontFamily -> "Times"], a1, {3, 0}], 
   Text[Style[B1, 12, FontFamily -> "Times"], b1, {-1, -2}], 
   Text[Style[C1, 12, FontFamily -> "Times"], c1, {0, 1}], 
   Text[Style[D1, 12, FontFamily -> "Times"], d1, {3, 0}]};
dashLines = {Dashed, 
   AbsoluteThickness[2], {Red, Line[{{c, d}, {c, c1}, {b, c}}]}};
realLines = {AbsoluteThickness[2], 
   Line[{{a, b}, {a, d}, {d, d1}, {b, b1}, {a1, a}, {b, b1}, {b1, 
      a1}, {b1, c1}, {c1, d1}, {a1, d1}}]};
Show[Graphics3D[{dashLines, realLines, labels}, Boxed -> False, 
  ViewPoint -> {2, 3.5, 1.28}], 
 Graphics3D[{Arrow[{{c1, c1 + {0, 0, 1}}, {d, d + {1, 0, 0}}, {b, 
      b + {0, 1, 0}}}], 
   Text[Style["z", 20, Italic, FontFamily -> "Times"], 
    c1 + {0, 0, 1}, {-1, -1}], 
   Text[Style["y", 20, Italic, FontFamily -> "Times"], 
    b + {0, 1, 0}, {-2, -1}], 
   Text[Style["x", 20, Italic, FontFamily -> "Times"], 
    d + {1, 0, 0}, {2, -1}]}]]

How can we determine if the four objects mentioned above can be placed in this cubic container with an edge length of one meter?

Solution:

Since the diameter of the Inscribed sphere of a cube with a 1 m edge length is 1 m, the first option is correct.

Since Regular tetrahedron with edge length of Sqrt [2] m can be placed in a cube with edge length of 1m, and Sqrt [2]>1.4, the second option is correct.

Due to the fact that the diagonal length of a cube with an edge length of 1m is Sqrt [3] m and Sqrt [3]<1.8, a cylinder with a height of 1.8m cannot be placed in a cube container. The third option is incorrect.

Since the length of the cube diagonal with the edge length of 1 m is Sqrt [3] m, and the height of 0.01 m of the cylinder with the bottom diameter of 1.2 m can be ignored, it is only necessary to place the bottom of the cylinder in parallel with the cube diagonal, that is, it can be placed in the cube container as a whole, so the fourth option is correct.

Answer 1

Here I present an approach which gives maximal sphere and maximal thetrahedron:

cube = Graphics3D[{FaceForm[None], EdgeForm[Blue], Cube[]} ,Boxed -> False]

sphere

sphere = NMaximize[{r, RegionWithin[Cube[], Sphere[{0, 0, 0}, r] ]}, r](*{0.5, {r -> 0.5}}*)
Show[{cube, Graphics3D[Sphere[{0, 0, 0}, r]]} /. sphere[[2]]]

Maximal radius r->0.5!

tetrahedron

pts = Table[{x[i], y[i], z[i]}, {i, 1, 4}];
tetra = NMaximize[{Volume[Tetrahedron[pts ]],RegionMember[cub, pts ] }, Flatten[pts]]
Show[cube, Graphics3D[{ Tetrahedron[pts ]} /. tetra[[2]]]]

Maximal sidelength 1.4142

Norm[pts[[1]] - pts[[2]]] /. tetra[[2]] (* Sqrt[2] *)