  In this paper we establish a framework for planar geometric interpolation
with exact area preservation using cubic B\'ezier polynomials. We show there
exists a family of such curves which are $5^{th}$ order accurate, one order
higher than standard geometric cubic Hermite interpolation. We prove this
result is valid when the curvature at the endpoints does not vanish, and in the
case of vanishing curvature, the interpolation is $4^{th}$ order accurate. The
method is computationally efficient and prescribes the parametrization speed at
endpoints through an explicit formula based on the given data. Additional
accuracy (i.e. same order but lower error constant) may be obtained through an
iterative process to find optimal parametrization speeds which further reduces
the error while still preserving the prescribed area exactly.
