Visualising Solid Shapes - Euler's Formula

Polyhedrons: These are 3D solid shapes bounded by polygons. Visually, a polyhedron is made up of flat surfaces called faces ($F$), straight line segments where two faces meet called edges ($E$), and corner points where edges meet called vertices ($V$). Examples include cubes and pyramids, whereas cylinders and spheres are not polyhedrons because they have curved surfaces.

Prisms: A prism is a polyhedron whose top and bottom (bases) are congruent polygons and whose side faces (lateral faces) are parallelograms or rectangles. Visually, if you slice a prism parallel to the base anywhere along its height, you get the same shape. A common example is a triangular prism which looks like a tent.

Pyramids: A pyramid is a polyhedron whose base is any polygon and whose lateral faces are triangles with a common vertex called the apex. Visually, it looks like a base on the ground with several triangles leaning inward to meet at a single point at the top, like the Great Pyramid of Giza.

Regular Polyhedrons: A polyhedron is called regular if its faces are made up of regular polygons (all sides and angles equal) and the same number of faces meet at each vertex. For example, a cube is a regular polyhedron because all $6$ faces are identical squares and exactly $3$ edges meet at every vertex.

Convex Polyhedrons: A polyhedron is convex if the line segment joining any two points on its surface lies entirely inside the solid. Visually, a convex polyhedron has no 'dents' or 'caves' in its structure; all its vertices point outwards.

Euler's Formula: This is a fundamental relationship for any convex polyhedron. It states that the number of faces ($F$) plus the number of vertices ($V$) minus the number of edges ($E$) always equals $2$. This formula allows us to find a missing dimension if the other two are known.

Dimensionality: Shapes can be $2D$ or $3D$. $2D$ shapes like squares and circles have only length and breadth, appearing flat on paper. $3D$ shapes like cubes and cones have length, breadth, and height (or depth), occupying space and having volume.