Laser cutters are 2D tools, but their speed and compatibility with a variety of affordable materials also makes them a frequent choice to create 3D objects. We propose CutCAD , a tool to easily construct simple 3D objects from 2D faces, inspired by the process of paper modeling and magnetic construction kits. The user creates her 3D model by drawing or loading existing 2D shapes, and connecting their edges in the software. CutCAD then automatically resolves the resulting constraints, and folds the faces up into a 3D model that is previewed live. CutCAD also automatically creates the required finger joints based on thickness of the material and dihedral angles, for smooth assembly. Cutouts are easy to add by importing their outlines as vector drawings, and placing them onto faces. After the faces have been cut, CutCAD provides assembly instructions. Observations and feedback from using CutCAD show the resulting process to be easier to understand than traditional 3D modelling. CutCAD is open-source, and has been downloaded over 2,000 times.