Geometry - Nets of 3D shapes

A net is a two-dimensional (2D) flat shape that can be folded along its edges to create a three-dimensional (3D) solid. Imagine taking a cardboard box and cutting along its edges to lay it completely flat on a table; the resulting pattern is the net of that box.

A Cube Net: Since a cube has $6$ identical square faces, its net must consist of exactly $6$ equal squares. A common visual for a cube net is the 'cross' shape, which has a vertical row of $4$ squares and $2$ squares attached to the left and right of the second square from the top. When folded, these squares meet to form $8$ vertices and $12$ edges.

A Cuboid Net: A cuboid (rectangular prism) has $6$ rectangular faces where opposite faces are equal. Its net consists of $6$ rectangles. Visually, these rectangles appear in $3$ matching pairs. If you fold the net, the rectangles of the same size will end up directly opposite each other to form the length ($l$), width ($w$), and height ($h$) of the cuboid.

A Cylinder Net: A cylinder has two flat circular bases and one curved surface. When 'unrolled', the curved surface becomes a large rectangle. Therefore, the net of a cylinder looks like a rectangle with two identical circles attached to the opposite long sides. The length of this rectangle is equal to the circumference of the circle.

A Cone Net: A cone has one circular base and a curved surface that tapers to a point (the apex). The net of a cone is made of a circle (the base) and a shape called a 'sector', which looks like a slice of a pie or a triangle with a curved bottom edge.

A Square-based Pyramid Net: This shape has $1$ square base and $4$ triangular faces. Its net typically looks like a central square with a triangle extending outward from each of the square's four sides, resembling a four-pointed star. When the triangles are folded upward, their tips meet at a single point called the vertex.

A Triangular Prism Net: A triangular prism consists of $2$ triangular bases and $3$ rectangular sides. The net usually shows a row of $3$ rectangles with $2$ triangles attached to the top and bottom of one of the rectangles. It has a total of $5$ faces.

Validating Nets: Not every arrangement of 2D shapes can form a 3D solid. For a net to be valid, the faces must not overlap when folded, and there must be no gaps. For example, a cube net must have exactly $6$ squares; if it has $5$ or $7$, it cannot form a perfect cube.