Net of a Solid
Geometry & MeasurementA 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 you unfold it and lay it flat. If you cut along some edges of a box and unfold it, you get a cross-shaped flat piece of cardboard. That flat shape is the net.
Example
A cube net has $6$ squares joined edge-to-edge. There are $11$ different ways to arrange $6$ squares that fold into a cube. The classic "plus sign" cross pattern is the most common.
Key Insight
Nets are used to calculate surface area: just add up the areas of all the flat shapes in the net. Packaging designers use nets to plan how boxes and cartons will be cut from flat sheets of cardboard.
Definition
A net is a two-dimensional pattern consisting of polygons joined along edges that, when folded, forms a closed 3-D polyhedron with no overlaps. Every polyhedron has at least one net (though some complex polyhedra's nets may be difficult to find). The surface area of the solid equals the area of its net.
Example
A triangular prism net has $2$ triangles and $3$ rectangles. If the triangular bases have legs $3$ and $4$ cm (hypotenuse $5$ cm) and the prism is $8$ cm long, the net area $= 2(1/2)(3)(4) + 3(8) + 4(8) + 5(8) = 12 + 24 + 32 + 40 = 108$ cm$^2$.
Key Insight
Not all arrangements of a shape's faces form valid nets. A cube has $11$ valid nets out of the $35$ possible hexominoes. Determining which arrangements fold correctly is an interesting combinatorial problem studied in computational origami.
Definition
A net is an edge-unfolding of a polyhedron: a spanning tree of the dual graph of the surface determines which edges are cut, and the remaining edges form fold lines. It is conjectured (Shephard, 1975) that every convex polyhedron has at least one non-overlapping net, but this remains unproven for all cases. Non-convex polyhedra may have no valid non-overlapping net.
Example
The $11$ nets of a cube correspond to the $11$ spanning trees of a path graph on $6$ nodes (up to symmetry). For a regular icosahedron ($20$ triangular faces), there are $43{,}380$ distinct nets. Enumerating nets is computationally intensive for large polyhedra.
Key Insight
Computational origami, developed by researchers including Erik Demaine, studies when and how nets can be folded into target shapes. The fold-and-cut theorem 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.