3D Shapes & Nets

A pyramid is a solid with a polygonal base and triangular faces that meet at a single point known as the apex.

The name of a pyramid is derived from the shape of its base. For example, a square-based pyramid has one square base and four triangular faces.

A tetrahedron is a special pyramid where the base is a triangle, resulting in a total of four triangular faces. If all four triangles are equilateral, it is a regular tetrahedron.

A cone is the circular equivalent of a pyramid. It has one circular base and a curved surface that tapers to an apex. In a net, this curved surface appears as a sector of a circle.

A net is a flat, 2D shape that can be folded along specific lines to create a 3D solid. It represents the 'unfolded' skin of the object.

The total area of a net is exactly equal to the total surface area of the 3D solid it forms. This makes nets a primary tool for calculating surface area in geometry.

For a cylinder, the net consists of two circles and one rectangle. Crucially, the length of this rectangle must be equal to the circumference of the circular base ($C = 2\pi r$).

A single 3D shape can often be represented by multiple different net configurations. For instance, there are 11 distinct ways to arrange six squares to form a valid net for a cube.

Face Counting: When identifying if a net is valid, first count the number of faces. A cube net must have exactly 6 squares; a triangular prism must have 2 triangles and 3 rectangles.

Edge Matching: Mentally fold the net to ensure that edges intended to meet are of equal length. If a rectangle side is meant to wrap around a circle, its length must be $2\pi r$.

Visualization: Imagine the base of the shape staying flat on the table and 'peeling' the other faces down. This helps in determining where faces should be attached in a net drawing.

Sanity Check: Ensure that no two faces in your net would overlap when folded. Overlapping faces indicate an invalid net arrangement.