Visualising Solid Shapes - Nets for Building 3-D Shapes

A Net is a 2-dimensional skeleton outline or flat pattern that can be folded to form a specific 3-dimensional solid shape. For example, if you unfold a cardboard shipping box and lay it flat, the resulting 2D shape is the net of a cuboid.

A net for a Cube consists of 6 identical squares. While there are 11 different possible nets for a cube, a common visual is a 'cross' shape featuring four squares in a vertical row and two squares attached to the sides of the second square from the top.

The net of a Cylinder is composed of one rectangle and two identical circles. When folded, the rectangle curves to form the lateral surface, and the two circles form the top and bottom bases. Visually, the length of the rectangle must equal the circumference of the circular bases ($2\pi r$).

A Cone's net consists of a sector of a circle (representing the curved lateral surface) and one small circle (representing the base). The arc length of the sector is exactly equal to the circumference of the base circle.

Prisms and Pyramids have distinct net structures. A Prism net has two identical polygonal bases and several rectangular lateral faces (e.g., a triangular prism has 2 triangles and 3 rectangles). A Pyramid net has one polygonal base with several triangular faces meeting at a single point or apex (e.g., a square pyramid net looks like a central square with four triangles attached to its edges).

Euler's Formula describes the relationship between the faces, vertices, and edges of any convex polyhedron. It states that the sum of the number of faces ($F$) and vertices ($V$) minus the number of edges ($E$) always equals 2.

The number of faces in a net must match the number of faces of the solid. For example, a tetrahedron (triangular pyramid) must have a net with exactly 4 triangles. Visually, these triangles must be arranged such that they share edges and can close to form a point without overlapping.