Geometry - Three-Dimensional Shapes: Cubes, Cuboids, Cylinders, Cones, Spheres

Understanding 3D Shapes: Three-dimensional shapes are solids that occupy space and have three dimensions: length, breadth (or width), and height. Unlike flat 2D shapes, these objects have volume and can be held. Common examples include boxes, balls, and cans.

Cuboid Properties: A cuboid is a solid bounded by six rectangular faces. It has 12 edges and 8 vertices. In a cuboid, opposite faces are congruent (identical). Visualize a matchbox or a brick; all the corners meet at right angles ($90^{\circ}$).

Cube Properties: A cube is a special case of a cuboid where all six faces are equal squares. This means the length, breadth, and height are all equal (denoted as side $a$). It also has 12 edges and 8 vertices. Imagine a standard playing die or a Rubik's cube.

Cylinder Characteristics: A cylinder consists of two congruent circular flat faces at the top and bottom, connected by a curved surface. It has 2 curved edges and 0 vertices. Visualize this as a tube or a standard soda can.

Cone Characteristics: A cone has one circular flat base that tapers smoothly to a single point called the apex or vertex. It has 1 curved surface and 1 curved edge. Visualize a funnel or a sharp pencil tip.

Sphere Characteristics: A sphere is a perfectly round 3D object where every point on its surface is equidistant from a fixed point called the center. It has 1 curved surface, 0 edges, and 0 vertices. Visualize a football or a marble.

Euler's Formula: For any polyhedron (a solid with flat faces and straight edges like cubes and cuboids), the relationship between the number of Faces ($F$), Vertices ($V$), and Edges ($E$) is given by the formula $F + V - E = 2$.

Surface Area and Volume: Volume is the measure of space occupied by a 3D object, measured in cubic units (like $cm^3$). Surface Area is the total area of all the faces of the solid, measured in square units (like $cm^2$).