  The tree amplituhedra $\mathcal{A}_{n,k}^{(m)}$ are mathematical objects
generalising the notion of polytopes into the Grassmannian. Proposed for $m=4$
as a geometric construction encoding tree-level scattering amplitudes in planar
$\mathcal{N}=4$ super Yang-Mills theory, they are mathematically interesting
for any $m$. In this paper we strengthen the relation between scattering
amplitudes and geometry by linking the amplituhedron to the Jeffrey-Kirwan
residue, a powerful concept in symplectic and algebraic geometry. We focus on a
particular class of amplituhedra in any dimension, namely cyclic polytopes, and
their even-dimensional conjugates. We show how the Jeffrey-Kirwan residue
prescription allows to extract the correct amplituhedron volume functions in
all these cases. Notably, this also naturally exposes the rich combinatorial
and geometric structures of amplituhedra, such as their regular triangulations.
