If
$f({{x}_{1}},...,{{x}_{k}})$
is a polynomial with complex coefficients, the Mahler measure of
$f$
,
$M(f)$
is defined to be the geometric mean of
$|f|$
over the
$k$
-torus
${{\mathbb{T}}^{k}}$
. We construct a sequence of approximations
${{M}_{n}}\,(f)$
which satisfy
$-d{{2}^{-n}}\,\log \,2\,+\,\log \,{{M}_{n}}(f)\,\le \,\log \,M(f)\,\le \,\log \,{{M}_{n}}(f)$
. We use these to prove that
$M(f)$
is a continuous function of the coefficients of
$f$
for polynomials of fixed total degree
$d$
. Since
${{M}_{n}}\,(f)$
can be computed in a finite number of arithmetic operations from the coefficients of
$f$
this also demonstrates an effective (but impractical) method for computing
$M(f)$
to arbitrary accuracy.