3D Solids

3D solids (three-dimensional solids) are geometric shapes that occupy space and have three dimensions: length, width, and height. These are foundational in geometry and play a significant role in mathematics, physics, and engineering.

Key Properties of 3D Solids

1. Volume:

   • The amount of space enclosed within the solid.

   • Measured in cubic units (e.g., cm³, m³).

2. Surface Area:

   • The total area of all external surfaces of the solid.

   • Measured in square units (e.g., cm², m²).

3. Faces:

   • Flat surfaces that make up the boundary of the solid.

4. Edges:

   • The line segments where two faces meet.

5. Vertices:

   • Points where edges meet.

Common Types of 3D Solids

1. Prisms:

   • Definition: Solids with two parallel, congruent bases connected by rectangular faces.

   • Examples: Rectangular prism, triangular prism, hexagonal prism.

2. Pyramids:

   • Definition: Solids with a polygonal base and triangular faces that meet at a single vertex.

   • Examples: Square pyramid, triangular pyramid.

3. Cylinders:

   • Definition: Solids with two parallel circular bases connected by a curved surface.

4. Cones:

   • Definition: Solids with a circular base and a curved surface that tapers to a point.

5. Spheres:

   • Definition: Solids where all points are equidistant from the center.

Euler's Formula for Polyhedra

For any convex polyhedron:

V−E+F=2V−E+F=2

Where:

• VV: Number of vertices.

• EE: Number of edges.

• FF: Number of faces.

Applications

• Architecture and Engineering: Understanding shapes like prisms and cylinders for structural designs.

• Physics: Calculating volume and surface area for fluid dynamics and material properties.

• 3D Modelling: Used in computer graphics for designing objects.