How big is my pizza, if I know its slices' sizes?

Let a pizza triangle be the convex envelope of the vertices of a pizza slice. The area of any pizza triangle is a bit smaller then the area of its pizza slice, but not so much (especially given the condition on the angles). It is easy to compute the area of a pizza triangle through the sine theorem, so a simple criterion is given by computing the sum of the areas of the pizza triangles and compare it with the area of a regular octagon inscribed in a circle with diameter $18$ inches.

Anyway, given two opposite pizza slices it is not difficult to compute the radius of the disk from which they have been cut, since two opposite pizza slices give a cyclic quadrilateral for which it is not difficult to compute the side lengths and the area given $a,b,c,d,\theta$:

hence the circumradius is provided by Parameshvara's formula:

$$ R = \frac{1}{4\Delta}\sqrt{(l_1 l_3+l_2 l_4)(l_1 l_2 + l_3 l_4)(l_1 l_4+l_2 l_3)} $$ where $l_1,l_2,l_3,l_4$ are the side lengths of the cyclic quadrilateral depicted above: they can be computed through the cosine theorem. Also notice that Ptolemy's theorem gives: $$l_1 l_3+l_2 l_4=(a+c)(b+d).$$

Another possible approach is the following. We have: $$\text{pow}_\Gamma(C) = ac = bd = R^2-OC^2, $$ so we just need to find $OC^2$. If we take $M$ and $N$ as the midpoints of the chords in the picture above, it is trivial that $OC$ is the diameter of the circumcircle of $CMN$, and we may compute the circumradius of $CMN$ through the sine theorem: $$\frac{OC}{2}=\frac{MN}{2\sin\theta}$$ then the length of $MN$ through the cosine theorem, so that: $$ OC^2 = \frac{1}{\sin^2\theta}\left(\left(\frac{a-c}{2}\right)^2+\left(\frac{b-d}{2}\right)^2-\frac{|a-c||b-d|}{2}\cos\theta\right) $$ and:

$$ R^2 = ac+\frac{1}{4\sin^2\theta}\left[(a-c)^2+(b-d)^2-2|a-c||b-d|\cos\theta\right].$$

If you do not know which couples of slices are "antipodal", well, they are not difficult to recognize: antipodal slices must have the same angle $\theta$ and fulfill $ac=bd$ (the intersecting chord theorem).