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A central composite design combines a factorial portion with axial points $\pm \alpha e_j$ and center runs. For a full $2^d$ factorial design, $d$ quantitative factors and $n_0\ge 1$ center runs, its run count is $2^d+2d+n_0$. The factorial portion supplies main effects and interaction terms, while the axial runs allow the individual squared terms of a full quadratic regression model to be estimated. Additional center replicates estimate pure error.

For two factors and $n_0=5$, this gives $4+4+5=13$ runs. The four corners are $(\pm 1,\pm 1)$, the four axial points are $(\pm \alpha ,0)$ and $(0,\pm \alpha )$, and the remaining five runs repeat $(0,0)$. Choosing $\alpha =\sqrt{2}$ makes this full-factorial design rotatable. Choosing $\alpha =1$ keeps the axial points on the faces and uses only three levels per factor, but does not give that rotatability in dimension at least two.

Circumscribed designs place the axial points outside the original factorial cube; inscribed designs rescale the cube inward to respect physical factor limits. A fractional factorial portion can reduce runs, but must retain sufficient resolution of a fractional factorial design to estimate the desired interactions: axial points have zero cross-products and cannot separate interactions that remain aliased in the factorial portion. The institutional handbook describes augmentation from a full or resolution V factorial portion.^[ref]