[asy][asy] import graph; size(4.662701220158751cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.337693339683693, xmax = 2.323308403400238, ymin = -0.39518100382374105, ymax = 4.44811779880594; /* image dimensions */ pen yfyfyf = rgb(0.5607843137254902,0.5607843137254902,0.5607843137254902); pen sqsqsq = rgb(0.12549019607843137,0.12549019607843137,0.12549019607843137); filldraw((-3.,4.)--(1.,4.)--(1.,0.)--(-3.,0.)--cycle, white, linewidth(1.6)); filldraw((0.,4.)--(-3.,2.)--(-2.,0.)--(1.,1.)--cycle, yfyfyf, linewidth(2.) + sqsqsq); /* draw figures */ draw((-3.,4.)--(1.,4.), linewidth(1.6)); draw((1.,4.)--(1.,0.), linewidth(1.6)); draw((1.,0.)--(-3.,0.), linewidth(1.6)); draw((-3.,0.)--(-3.,4.), linewidth(1.6)); draw((0.,4.)--(-3.,2.), linewidth(2.) + sqsqsq); draw((-3.,2.)--(-2.,0.), linewidth(2.) + sqsqsq); draw((-2.,0.)--(1.,1.), linewidth(2.) + sqsqsq); draw((1.,1.)--(0.,4.), linewidth(2.) + sqsqsq); label("$A$",(-3.434705427329005,4.459844914550807),SE*labelscalefactor,fontsize(14)); label("$a$",(-1.634705427329005,4.459844914550807),SE*labelscalefactor,fontsize(14)); label("$B$",(1.056779902954702,4.424663567316209),SE*labelscalefactor,fontsize(14)); label("$b$",(-3.456779902954702,3.24663567316209),SE*labelscalefactor,fontsize(14)); label("$C$",(1.0450527872098359,0.07390362597090126),SE*labelscalefactor,fontsize(14)); label("$c$",(1.250527872098359,0.7390362597090126),SE*labelscalefactor,fontsize(14)); label("$D$",(-3.551976584777666,0.12081208895036549),SE*labelscalefactor,fontsize(14)); label("$d$",(-.551976584777666,0.09081208895036549),SE*labelscalefactor,fontsize(14)); /* dots and labels */ dot((-2.,4.),linewidth(4.pt) + dotstyle); dot((-1.,4.),linewidth(4.pt) + dotstyle); dot((0.,4.),linewidth(4.pt) + dotstyle); dot((1.,3.),linewidth(4.pt) + dotstyle); dot((1.,2.),linewidth(4.pt) + dotstyle); dot((1.,1.),linewidth(4.pt) + dotstyle); dot((0.,0.),linewidth(4.pt) + dotstyle); dot((-1.,0.),linewidth(4.pt) + dotstyle); dot((-3.,1.),linewidth(4.pt) + dotstyle); dot((-3.,2.),linewidth(4.pt) + dotstyle); dot((-3.,3.),linewidth(4.pt) + dotstyle); dot((-2.,0.),linewidth(4.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy] Let $a,b,c,d$ be the lengths of the sides. Then $4-a,4-b,4-c,4-d$ are the lengths of the complementary sides. To find the area, we can use complementary areas. So we can find the area of the triangles and subtract. $\frac{ab+(4-a)(4-b)+(4-b)(4-d)+cd}{2}$ This ugly expression can be factored to: $\frac{16+(a+b-4)(b+c-4)}{2}$ Subtracting this from the orginal gives $8-(a+d-4)(b+c-4)$ Since $a,b,c,d$ are from $[1,3]$, then the value of $a+b$ or $c+d$ can be $2,3,4,5,6$ We have to try all those combinations in the equation. We substitute $a+b$ and $c+d$ for all combos of $2,3,4,5,6$ It can be: $(2,2),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(6,6)$ The value of $(a+d-4)(b+c-4)$ can thus be $-2,-1,-.5,0,.5,1.2$ Then $8$ minus all of those numbers gives: $\boxed{6,7,7.5,8,8.5,9,10}$

Let E,F,G,H be the vertices of the quadrilateral on AB,BC,CD, and AD respectively. You can see that the area of the quadrilateral is $\frac{16-(BF-AH)(AE-DG)}2$. The possible values for$BF-AH$ or $AE-DG$ are -2,-1,0,1,2. Now plug in the numbers to get 6,7,7.5,8,8.5,9,10. [asy][asy] import graph; size(4.662701220158751cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.337693339683693, xmax = 2.323308403400238, ymin = -0.39518100382374105, ymax = 4.44811779880594; /* image dimensions */ pen yfyfyf = rgb(0.5607843137254902,0.5607843137254902,0.5607843137254902); pen sqsqsq = rgb(0.12549019607843137,0.12549019607843137,0.12549019607843137); filldraw((-3.,4.)--(1.,4.)--(1.,0.)--(-3.,0.)--cycle, white, linewidth(1.6)); filldraw((0.,4.)--(-3.,2.)--(-2.,0.)--(1.,1.)--cycle, yfyfyf, linewidth(2.) + sqsqsq); /* draw figures */ draw((-3.,4.)--(1.,4.), linewidth(1.6)); draw((1.,4.)--(1.,0.), linewidth(1.6)); draw((1.,0.)--(-3.,0.), linewidth(1.6)); draw((-3.,0.)--(-3.,4.), linewidth(1.6)); draw((-3.,2.)--(0.,2.), linewidth(1.6)); draw((-2.,1.)--(1.,1.), linewidth(1.6)); draw((-2.,2.)--(-2.,0.), linewidth(1.6)); draw((-0.,4.)--(0.,1.), linewidth(1.6)); draw((-3.,4.)--(-3.,2.), linewidth(2.) + sqsqsq); draw((-3.,2.)--(-2.,0.), linewidth(2.) + sqsqsq); draw((-2.,0.)--(1.,1.), linewidth(2.) + sqsqsq); draw((1.,1.)--(0.,4.), linewidth(2.) + sqsqsq); label("$A$",(-3.434705427329005,4.459844914550807),SE*labelscalefactor,fontsize(14)); label("$B$",(1.056779902954702,4.424663567316209),SE*labelscalefactor,fontsize(14)); label("$C$",(1.0450527872098359,0.07390362597090126),SE*labelscalefactor,fontsize(14)); label("$D$",(-3.551976584777666,0.12081208895036549),SE*labelscalefactor,fontsize(14)); /* dots and labels */ dot((-2.,4.),linewidth(4.pt) + dotstyle); dot((-1.,4.),linewidth(4.pt) + dotstyle); dot((0.,4.),linewidth(4.pt) + dotstyle); dot((1.,3.),linewidth(4.pt) + dotstyle); dot((1.,2.),linewidth(4.pt) + dotstyle); dot((1.,1.),linewidth(4.pt) + dotstyle); dot((0.,0.),linewidth(4.pt) + dotstyle); dot((-1.,0.),linewidth(4.pt) + dotstyle); dot((-3.,1.),linewidth(4.pt) + dotstyle); dot((-3.,2.),linewidth(4.pt) + dotstyle); dot((-3.,3.),linewidth(4.pt) + dotstyle); dot((-2.,0.),linewidth(4.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy]