I use conformal mapping techniques to determine the change in the conductivity of a sheet containing a few well-separated holes. The hole shapes studied are the equilateral triangle, square, pentagon and regular
n
-gons. I show that the conductivity can be written as
σ
/
σ
0
= 1 –
α
n
f
+
o
(
f
2
), where
f
is the area fraction of the inclusions and the coefficient
α
n
= (tan (π/
n
)/2π
n
) Г
4
(1/
n
)/Г
2
(2/
n
), which is 2.5811, 2.1884, 2.0878 for triangles, squares and pentagons, and tends to the circle limit of 2 as
n
→∞ . The coefficient α
n
is proportional to the induced dipole moment around the polygonal hole which can be found using an appropriate conformal mapping. I have also examined and compared the results for long thin needle-like holes in the shape of diamonds, rectangles and ellipses. In all cases the conductivity parallel to the needles has the limiting form
σ
/
σ
0
= 1 –
f
, while for the perpendicular conductivity, I find that
σ
/
σ
0
= 1 –
n
π
a
2
, where 2
a
is the length of the needle, and
n
is the number of needles per unit area. For thicker needles, the shape becomes important and I compare the results with recent analog experiments and computer simulations. Because of the reciprocity theorem, all the results found here apply equally well to superconducting inclusions.
