Net of a Solid
A net of a solid is a flat 2-D pattern that can be folded along its edges to form the 3-D solid.
Definition
A net is what a 3-D shape looks like when unfolded and laid flat: a two-dimensional pattern of polygons joined along edges that, when folded, forms the closed solid with no overlaps, and the surface area of the solid equals the area of its net. Every polyhedron has at least one net, though complex polyhedra's nets can be hard to find; a cube has $11$ valid nets out of $35$ possible hexomino arrangements. Formally, a net is an edge-unfolding of a polyhedron, determined by a spanning tree of the dual graph of the surface (the cut edges), with the remaining edges forming fold lines; it is conjectured (Shephard, 1975) that every convex polyhedron has at least one non-overlapping net, but this remains unproven in general, and non-convex polyhedra may have no valid non-overlapping net at all.
Example
If you cut along some edges of a box and unfold it, you get a cross-shaped flat piece of cardboard, the net; a cube net has $6$ squares joined edge-to-edge, and the classic "plus sign" cross is the most common of the $11$ valid arrangements. A triangular prism net with legs $3$ and $4$ cm (hypotenuse $5$ cm) and length $8$ cm has net area $= 2(1/2)(3)(4) + 3(8) + 4(8) + 5(8) = 12 + 24 + 32 + 40 = 108$ cm$^2$. A regular icosahedron ($20$ triangular faces) has $43{,}380$ distinct nets, making enumeration computationally intensive for large polyhedra.
Key Insight
Nets are used to calculate surface area by simply adding up the areas of all the flat shapes, and packaging designers use them to plan how boxes and cartons will be cut from flat sheets of cardboard. Determining which arrangements of a shape's faces fold correctly is a combinatorial problem studied in computational origami; the fold-and-cut theorem, developed by researchers including Erik Demaine, states that any straight-line figure can be created by folding paper flat and making a single straight cut, a surprising result with deep geometric content.