Unit 19 Lesson 1: 3D Shapes - Faces or Sides

Three-Dimensional (3D) Geometry Fundamentals

• Three Dimensions Defined:

  • 3D shapes possess three spatial dimensions: length (height), width (breadth), and depth.
  • They are solid shapes enclosing a volume of space.

• Fundamental Surface and Boundary Components:

  • Faces (or Sides): Flat or curved surfaces forming the exterior boundary of a solid. All 3D shapes have faces.
  • Edges: Straight or curved boundary lines formed where two faces meet. All 3D shapes have edges apart from a sphere.
  • Vertices (or Corners): Pointed intersections where three or more edges converge. The plural form is vertices; the singular form is vertex. All 3D shapes have vertices apart from a sphere.

Geometric Definitions: Faces, Edges, Vertices, and Polyhedra

• Polyhedron Definition:

  • A solid geometric figure in three dimensions bounded entirely by flat polygon faces.
  • In any polyhedron, faces meet at straight line segments called edges, and edges meet at sharp corner points called vertices.
  • Solids possessing any curved surfaces (such as cylinders, cones, and spheres) are strictly classified as non-polyhedra.

• Face Geometry:

  • The individual faces of polyhedra are specific two-dimensional geometric polygons (such as squares, rectangles, triangles, pentagons, or hexagons).

Prisms: Characteristics and Properties

• Definition of a Prism:

  • A 3D solid shape where all faces are completely flat.
  • It features two identical, parallel polygonal faces situated at opposite ends, designated as the bases.
  • It possesses a uniform (constant) cross-section throughout its entire length when sliced parallel to its bases.
  • The geometric shape of the parallel end faces defines the specific name of the prism.

• Classification of Prisms:

  • Cube:
  • A specialized prism bounded entirely by $6$ congruent square-shaped faces.
  • Number of faces: $6$
  • Number of edges: $12$
  • Number of vertices: $8$
  • Cuboid (Rectangular Prism):
  • A prism with rectangular faces (and optionally two square opposite faces).
  • It maintains an identical rectangular cross-section along its entire length.
  • Number of faces: $6$
  • Number of edges: $12$
  • Number of vertices: $8$
  • Triangular Prism:
  • A prism bounded by $2$ identical parallel triangular bases connected by $3$ rectangular side faces.
  • Number of faces: $5$ (consisting of $2$ triangular faces and $3$ rectangular faces)
  • Number of edges: $9$
  • Number of vertices: $6$
  • Pentagonal Prism:
  • A prism bounded by $2$ identical parallel regular or irregular pentagonal bases connected by $5$ rectangular side faces.
  • Number of faces: $7$
  • Number of edges: $15$
  • Number of vertices: $10$

• Hexagonal Prism:
  • A prism bounded by $2$ identical parallel hexagonal bases connected by $6$ rectangular side faces.
  • Number of faces: $8$ (consisting of $2$ hexagonal faces and $6$ rectangular faces)
  • Number of edges: $18$
  • Number of vertices: $12$

Pyramids: Characteristics and Properties

• Definition of a Pyramid:

  • A polyhedron formed by connecting a polygon-shaped base to a single topmost point called the apex.
  • The lateral faces extending from the base perimeter up to the apex are always triangles.
  • A pyramid is formally classified and named according to the geometric shape of its polygon base.

• Key Types of Pyramids:

  • Square-Based Pyramid:
  • Base polygon: $1$ square.
  • Lateral faces: $4$ triangular faces converging at the apex.
  • Total faces: $5$
  • Total edges: $8$
  • Total vertices: $5$ ($4$ base vertices plus $1$ apex)
  • Triangular-Based Pyramid (Tetrahedron):
  • Base polygon: $1$ triangle.
  • Lateral faces: $3$ triangular faces converging at the apex.
  • Total faces: $4$ triangular faces.
  • Total edges: $6$
  • Total vertices: $4$
  • Regular Tetrahedron: A specialized triangular-based pyramid where all $4$ faces are congruent equilateral triangles.

Non-Polyhedral Solids (Curved 3D Shapes)

• General Properties:

  • Three-dimensional solid objects containing at least one curved boundary surface. Because their surfaces are not composed exclusively of planar polygons, they do not belong to the family of polyhedra.

• Cylinder:

  • Bounded by $2$ flat parallel circular base faces and $1$ curved lateral surface extending between them.
  • Total faces: $3$ ($2$ flat circular faces and $1$ curved face)
  • Total edges: $2$ circular curved edges
  • Total vertices: $0$

• Cone:

  • Bounded by $1$ flat circular base face and $1$ curved conical surface tapering to a single point (apex).
  • Total faces: $2$ ($1$ flat circular face and $1$ curved face)
  • Total edges: $1$ circular curved edge
  • Total vertices: $1$ vertex (the apex)

• Sphere:

  • A completely symmetrical, continuous curved solid surface everywhere equidistant from a single central point.
  • Total faces: $1$ continuously curved surface
  • Total edges: $0$
  • Total vertices: $0$

• Hemisphere:

  • Exactly half of a sphere, created by a planar cross-section passing through the sphere's geometric center.
  • Total faces: $2$ ($1$ flat circular base face and $1$ curved hemispherical dome face)
  • Total edges: $1$ circular curved edge
  • Total vertices: $0$

Reference Properties of Standard 3D Shapes

• Comprehensive Property Table:

• Key Mathematical Vocabulary:
  • face
  • edge
  • vertex / vertices
  • prism
  • cube
  • cuboid
  • tetrahedron
  • compound 3D shape
  • component shape

Geometric Reasoning and Nets

• Definition of a Net:
  • A two-dimensional pattern or unfolded flattened diagram of an open three-dimensional solid.
  • When folded along marked edges, the net forms the complete exterior boundary of the 3D solid without overlaps.
  • A net of an everyday object (such as a cardboard box) can be observed directly by carefully detaching joints and opening the container flat onto a single plane.

Compound 3D Shapes and Component Decomposition

• Definition of a Compound 3D Shape:

  • A complex structural shape formed by combining two or more distinct component 3D shapes.
  • Any internal faces where two component shapes are joined together cease to be exterior faces of the resulting compound solid.

• Decomposition Strategy:

  • To analyze or describe a compound 3D shape, deconstruct the overall structure into simpler, known 3D constituent shapes (e.g., decomposing a stepped L-block into a cube and a cuboid, or into two cuboids).

• "The House Year 6 Built" Structural Analysis:

  • Base Structure: A cuboid (rectangular prism).
  • Roof Structure: A triangular prism.
  • Shared Property: Both the base and roof components are prisms because each has a uniform cross-section along its length.

Quantitative Analysis of Compound 3D Structures

• Specific Compound Assemblies and Their Total Exterior Counts:

  • Cylinder and Cone Assembly (Joined at circular base):

  • Component shapes: $1$ cylinder and $1$ cone.

  • Faces: $2$ (the bottom flat circular base and the combined curved lateral surfaces).

  • Vertices: $1$ vertex (the apex of the cone).

  • Edges: $2$ (the bottom circular perimeter edge and the circular joint edge).

  • Cone on Top of a Hemisphere (Joined at circular base):

  • Component shapes: $1$ cone and $1$ hemisphere.

  • Faces: $2$ curved faces (conical curved surface + hemispherical curved surface).

  • Vertices: $1$ vertex (the apex of the cone).

  • Edges: $1$ circular edge (at the boundary seam where the two shapes meet).

  • **Cuboid and Square-Based Pyramid (Equal