Bake a slice of Pi

MATL, 70 68 67 bytes

'()'12:)l10:&<toYP43Y$51hb(!10Xy'\::\'FFhZ++'|'3$Yc'||\'3:(95'Zd'o(

Try it online!

Explanation

What a mess. But hey, there's a convolution!

The explanation will be clearer if you can inspect the stack contents after a given statement. To do it, just insert X#0$% at that point. (This means: X# show stack contents, 0$ don't implicitly display anything else, % comment out rest of the code). For example, see the stack right after the convolution.

'()'       % Push this string
12:        % Range [1 2 ... 12]
)          % Index into string (modular, 1-based): gives '()()()()()()'
l          % Push 1 (will be used later)
10:        % Range [1 2 ... 10]
&<         % All pairwise "less than" comparisons. Gives matrix with "true"
           % below the main diagonal, and the remining entries equal to "false"
to         % Duplicate. Convert to numbers (true becomes 1, false becomes 0)
YP43Y$     % Compute pi with 43 significant digits (42 decimals). Gives a string
51h        % Append last decimal, '3' (ASCII 51). This is needed to avoid rounding
b          % Bubble up the true-false matrix, to be used as logical index
(          % Fill the chars from the pi string into the 0-1 matrix, at the positions
           % indicated by the true-false matrix. Thus each 1 is replaced by a char
           % from the pi string. Entries that were 0 remain as 0. This is done in
           % columm-major order...
!          % ...so transpose to make it row-major
10Xy       % Identity matrix of size 10
'\::\'     % Push this string...
FFh        % ...and append two zeros
Z+         % 2D convolution keeping size. The identity matrix convolved with the
           % above string gives the diagonal bands with chars '\'  and ':'
+          % Add to the matrix containing the digits of pi. At each entry, only one
           % of the two matrices is nonzero
'|'        % Push this string
3$Yc       % Three-input string concatenation. This prepends the 1 (which was pushed
           % a while ago) and appends '|' to each row of the matrix. This converts
           % the matrix to char. Note that char 1 will be displayed as a space. We
           % used char 1 and not char 0 (which would be displayed as a space too)
           % because function `Yc` (`strcat`) strips  off trailing space from the
           % inputs, counting char 0 as space, but not char 1
'||\'      % Push this string
3:(        % Assign it to the first 3 entries of the matrix (column-major), that is, 
           % to the top of the first column
95         % Push ASCII for '_'
'Zd'o      % Push string 'Zd' and convert to numbers: gives [90 100]. These are the
           % (column-major) indices where the '_' char should appear in the last row
(          % Fill those chars
           % Implicitly display. (Chars 0 and 1 are displayed as space)

Perl, 93 bytes

$_=bpi$=;printf'()'x6x!$`.'
%12s',F.ee x!$\--^substr"\32::\\$&|",-12while/.{$\}/g

Requires the command line option -l71Mbignum=bpi, counted as 14. The \32 should be replaced by a literal character 26.

Sample Usage

$ perl -l71Mbignum=bpi pi-slice.pl
()()()()()()
|\3.1415926|
|:\53589793|
\::\2384626|
 \::\433832|
  \::\79502|
   \::\8841|
    \::\971|
     \::\69|
      \::\3|
       \__\|

Perl, 111 bytes

$_=bpi$_*($l=($.=$_)-3);printf'()'x($./2)x!$`."
%$.s",F.ee x!$l--^substr"\32::\\$&|",-$.while/.{$l}/g

Parameterized version. Requires the command line option -nMbignum=bpi, counted as 12.

Sample Usage

$ echo 10 | perl -nMbignum=bpi pi-slice.pl
()()()()()
|\3.14159|
|:\265358|
\::\97932|
 \::\3846|
  \::\264|
   \::\33|
    \::\8|
     \__\|

$ echo 20 | perl -nMbignum=bpi pi-slice.pl
()()()()()()()()()()
|\3.141592653589793|
|:\2384626433832795|
\::\028841971693993|
 \::\75105820974944|
  \::\5923078164062|
   \::\862089986280|
    \::\34825342117|
     \::\0679821480|
      \::\865132823|
       \::\06647093|
        \::\8446095|
         \::\505822|
          \::\31725|
           \::\3594|
            \::\081|
             \::\28|
              \::\4|
               \__\|

JavaScript (ES6), 187 174 bytes

This is 1 byte shorter than just displaying the plain text.

for(y=n=0,s=`()()()()()()
`;y<10;y++,s+=`|
`)for(x=-2;x++<9;)s+=x>y?(Math.PI+'2384626433832795028841971693')[n++]:`\\${y>8?'__':x+1|y>2?'::':'||'}\\`[y-x]||' ';console.log(s)