Let \(P \subseteq M_\RR\) be a full-dimensional very ample lattice polytope of dimension \(n\), embedded by the lattice points of \(P\). Then \[ \deg X_P = n! \cdot \vol(P) , \] the volume normalised so that the standard simplex has volume \(1/n!\). Equivalently \(\deg X_P = D_P^{\,n}\), the top self-intersection of the ample divisor.
(Ehrhart, in one line)
The statement is the leading term of the lattice-point count. Since \(h^0(X_P, \OO(kD_P)) = \size(kP \intersect M)\) and Ehrhart’s theorem makes this a polynomial in \(k\) of degree \(n\) with leading coefficient \(\vol(P)\), comparison with the Hilbert polynomial of \(X_P\) gives the degree. So the same lattice-point count that computes sections also computes the degree; only the leading coefficient is read instead of the value at \(k = 1\).
(Basic checks)
For \(P = \Delta\), the standard simplex, \(\vol = 1/n!\) and \(\deg \PP^n = 1\). For \(P = d\Delta\), \(\vol = d^n/n!\) and \(\deg = d^n\), which is the degree of the \(d\)-uple Veronese image of \(\PP^n\). For the unit square, \(2! \cdot 1 = 2\), the degree of the quadric surface \(\PP^1 \times \PP^1 \subseteq \PP^3\).