Constants with formulae of the form treated by D. Bailey, P. Borwein, and S. Plouffe (
$b$
formulae to a given base
$b$
) have interesting computational properties, such as allowing single digits in their base
$b$
expansion to be independently computed, and there are hints that they should be normal numbers, i.e., that their base
$b$
digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call Machin-type BBP formulae, for which it is relatively easy to determine whether or not a given constant
$K$
has a Machin-type BBP formula. In particular, given
$b\,\in \,\mathbb{N},\,b\,>\,2,\,b$
not a proper power, a
$b$
-ary Machin-type BBP arctangent formula for
$K$
is a formula of the form
$k\,=\,{{\Sigma }_{m}}\,{{a}_{m}}\,\arctan \,(-{{b}^{-m}}),\,{{a}_{m}}\,\in \,\mathbb{Q}$
, while when
$b\,=\,2$
, we also allow terms of the form
${{a}_{m}}\,\arctan \,(1/1\,-\,{{2}^{m}}))$
. Of particular interest, we show that
$\pi$
has no Machin-type BBP arctangent formula when
$b\,\ne \,2$
. To the best of our knowledge, when there is no Machin-type BBP formula for a constant then no BBP formula of any form is known for that constant.