How to get all definitions associated with other symbols?

Question

How to get all definitions for a symbol associated with other symbols by TagSet, TagSetDelayed, UpSet or UpSetDelayed?

For example, if one has defined

area[square] ^= s^2
area[cube] ^= 6*s^2

how to obtain these definitions, not knowing the names square, cube but knowing only the name area?

---

I just have found that UpValues does not return definitions for MakeBoxes and N since they are stored in FormatValues and NValues correspondingly:

In[1]:= rotate /: MakeBoxes[expr_rotate, "StandardForm"] := x
UpValues[rotate]
FormatValues[rotate]

Out[2]= {}

Out[3]= {HoldPattern[MakeBoxes[expr_rotate, "StandardForm"]] :> x}

In[4]:= pi /: N[pi] = 3.14
UpValues[pi]
NValues[pi]

Out[4]= 3.14

Out[5]= {}

Out[6]= {HoldPattern[N[pi, {MachinePrecision, MachinePrecision}]] :> 
  3.14}

In this way instead of UpValues we should use a combination of UpValues, FormatValues and NValues.

---

When trying to output a list of FormatValues one can face problems with MakeBoxes since FormatValues gives definitions for MakeBoxes those are further processed by MakeBoxes on creating the output for the FrontEnd. This problem can be solved by switching FormatType temporarily to OutputForm or by converting these definitions to strings.

In[1]:= SetOptions[$Output,FormatType->OutputForm];
FormatValues[DialogNotebook]
Out[2]= {HoldPattern[MakeBoxes[BoxForm`apat$:HoldPattern[DialogNotebook[___]], BoxForm`fpat$_]] :> 

   BoxForm`BoxFormAutoLoad[MakeBoxes, BoxForm`apat$, BoxForm`fpat$, Typeset`CellNotebook`, 

    {{CellGroup, _}, {DocumentNotebook, _}, {PaletteNotebook, _}, {DialogNotebook, _}, {ExpressionCell, _}, {Text, _}, 

     {TextCell, _}, {Cell, HoldPattern[MakeExpression[_Cell, _]]}, {Notebook, HoldPattern[MakeExpression[_Notebook, _]]}}]}

In[1]:= ToString@FormatValues[DialogNotebook]
Out[1]= {HoldPattern[MakeBoxes[BoxForm`apat$:HoldPattern[DialogNotebook[___]], BoxForm`fpat$_]] :> BoxForm`BoxFormAutoLoad[MakeBoxes, BoxForm`apat$, BoxForm`fpat$, Typeset`CellNotebook`, {{CellGroup, _}, {DocumentNotebook, _}, {PaletteNotebook, _}, {DialogNotebook, _}, {ExpressionCell, _}, {Text, _}, {TextCell, _}, {Cell, HoldPattern[MakeExpression[_Cell, _]]}, {Notebook, HoldPattern[MakeExpression[_Notebook, _]]}}]}

Answer 1

Attempting to address Alexey's concerns with Howard's answer, I came up with this:

Cases[
   UpValues @@@ MakeExpression /@ Names["Global`*"],
   HoldPattern[_@_area :> _],
   {2}
]

---

In response to your updated requirements, here is the advanced version:

SetAttributes[otherValues, HoldFirst]

otherValues[sym_] :=
  With[{names = MakeExpression /@ Names["Global`*"]},
    Join[
      Cases[UpValues @@@ names, HoldPattern[_@_sym :> _], {2}],
      Cases[NValues @@@ names, HoldPattern[_@N[sym, ___] :> _], {2}],
      Select[Join @@ FormatValues @@@ names, ! FreeQ[#, HoldPattern@sym] &]
    ]
  ]