7.2 3D Solids, Nets, Faces/Edges/Vertices, and Orthographic Projections

Classifying 3D Solids: Prisms, Pyramids, and Curved Solids

Spatial reasoning items frequently test the structural properties of three-dimensional objects used in geometry models, packaging, and school architecture. Polyhedra (solids bounded entirely by flat polygon faces) fall into two main families:

1. Prisms: Have two congruent, parallel polygon bases (giving a uniform cross-section along their length) connected by rectangular lateral faces. A prism is named after the shape of its polygon base (e.g., a triangular prism has two triangular ends and three rectangular sides). For a prism whose base is an $n$-sided polygon:

   • Faces: $F = n + 2$ ($2$ bases plus $n$ rectangular lateral faces)
   • Vertices (corners): $V = 2n$ ($n$ vertices on the top base and $n$ on the bottom base)
   • Edges: $E = 3n$ ($n$ edges on the top base, $n$ on the bottom base, and $n$ vertical/lateral edges connecting them)

2. Pyramids: Have one polygon base and triangular lateral faces that slant upward to meet at a single point called the apex. For a pyramid whose base is an $n$-sided polygon:

   • Faces: $F = n + 1$ ($1$ base plus $n$ triangular lateral faces)
   • Vertices: $V = n + 1$ ($n$ base vertices plus $1$ apex)
   • Edges: $E = 2n$ ($n$ base edges plus $n$ slanted edges meeting at the apex)

3. Curved 3D Solids (Non-Polyhedra):

   • Cylinder: Bounded by $2$ parallel circular flat faces and $1$ curved lateral surface. Although it has a uniform circular cross-section like a prism, it is not a polyhedron because its wall is curved.
   • Cone: Bounded by $1$ circular flat base and $1$ curved surface tapering to $1$ apex.
   • Sphere: Bounded by $1$ continuous curved surface where every surface point is equidistant from the centre; it has $0$ flat faces, $0$ edges, and $0$ vertices.

Euler's Formula for Polyhedra

For any convex polyhedron (every prism and pyramid), the number of faces ($F$), vertices ($V$), and edges ($E$) satisfies Euler's formula:

$$F + V - E = 2 \quad \Longleftrightarrow \quad F + V = E + 2$$

For example, if a polyhedron has $7$ faces and $10$ vertices, you can immediately find its number of edges without visualising the entire solid: $E = F + V - 2 = 7 + 10 - 2 = 15$ edges (which corresponds to a pentagonal prism, where $n = 5$).

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2D Nets and Tracking Opposite Faces on a Cube

A net is a flat 2D pattern of polygons joined along shared edges that can be folded along its interior lines to enclose a 3D solid without any overlapping or missing faces.

• Prism nets: Consist of a strip of $n$ rectangles (which wrap around the lateral sides) with the two congruent $n$-gon bases attached to opposite sides of the rectangular strip. If both end polygons are attached to the same side of the strip, they will fold onto the same end and leave the opposite end open.
• Cylinder and cone nets: A closed cylinder's net consists of $1$ central rectangle (whose width equals the circle's circumference $2\pi r$) and $2$ circles attached to opposite long edges. A closed cone's net consists of $1$ circle attached to the curved arc of $1$ circular sector (a wedge of a larger circle).
• Pyramid nets: Consist of the single base polygon surrounded by $n$ triangles (typically arranged like a star around the base edges).
• Cube nets: There are $11$ valid nets composed of $6$ congruent squares:
  • $1\text{–}4\text{–}1$ family ($6$ nets): A central row of $4$ squares (which wrap into a hollow tube) with $1$ square on the top side and $1$ square on the bottom side (forming the two end caps).
  • $2\text{–}3\text{–}1$ family ($3$ nets): Rows of $2$, $3$, and $1$ squares.
  • $2\text{–}2\text{–}2$ staircase ($1$ net) and $3\text{–}3$ step ($1$ net).
  • Common invalid traps: Any arrangement containing $5$ or $6$ squares in a single straight line, any arrangement containing a $2 \times 2$ block of four squares, or any $1\text{–}4\text{–}1$ pattern where both end flaps sit on the same side of the $4$-square spine.

The Skip-One Rule for Opposite Faces on a Cube Net

Many spatial questions show a labelled net of a cube (with numbers, letters, or symbols on each square) and ask either which face is opposite a given face or which 3D perspective drawing is valid.

1. Rule 1 (Straight-line skip-one): Whenever three or more squares lie in a straight row or column on a net, two squares separated by exactly one square between them ($A - B - C$) fold to become opposite faces ($A$ is opposite $C$).
2. Rule 2 (Opposite faces never touch): Two opposite faces can never share an edge or be visible together in a standard three-face corner view of a cube.
3. Rule 3 (Process of elimination): Once you identify two pairs of opposite faces using the skip-one rule, the remaining two squares on the net are automatically the third pair of opposite faces.

Worked Example 1: Finding Opposite Faces on a Classroom Dice Net

A cube net is arranged in a cross ($1\text{–}4\text{–}1$): the horizontal row of four squares is labelled P, Q, R, S from left to right; a square labelled T is attached above Q, and a square labelled U is attached below R.

1. Along the horizontal row P – Q – R – S, skipping one square shows P is opposite R and Q is opposite S.
2. The two remaining squares are T and U, so T must be opposite U.
3. Therefore, if a 3D drawing shows faces P and R adjacent to each other, that drawing is impossible.

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Orthographic Views and Counting Stacked Cubes

Orthographic Projections (Plan and Elevations)

Architects and design teachers represent 3D structures using flat 2D orthographic views taken from perpendicular lines of sight:

• Top view (Plan view): Looking straight down from above. It shows which grid squares of the base footprint are occupied.
• Front view (Front elevation): Looking horizontally from the front. Each column's height equals the tallest stack in that left-to-right column.
• Back view: Looking horizontally from behind. Its silhouette is the horizontal mirror image (left-to-right reflection) of the front view.
• Left side view and Right side view: Looking horizontally from the left or right. Each column's height equals the tallest stack in that front-to-back row, and the left and right views are horizontal mirror images of each other.

Worked Example 2: Elevations and Total Cube Count from a Plan Grid

A block tower sits on a $2 \times 2$ base grid. Looking from above (top view), the column heights are: Front-Left $= 1$, Front-Right $= 3$, Back-Left $= 2$, and Back-Right $= 4$. What are the column heights seen in the Front view (from left to right) and the Left side view (from front to back), and how many unit cubes are in the tower?

1. Front view (left column vs. right column): The left column has heights $1$ (front) and $2$ (back), so its maximum height is $\max(1, 2) = 2$. The right column has heights $3$ (front) and $4$ (back), so its maximum height is $\max(3, 4) = 4$. Thus the Front view shows columns of height $[2, 4]$ from left to right.
2. Left side view (back row on left vs. front row on right): Looking from the left side of the model, the back row is on the viewer's left with maximum height $\max(2, 4) = 4$, and the front row is on the viewer's right with maximum height $\max(1, 3) = 3$.
3. Total unit cubes: Sum the four vertical stacks: $1 + 3 + 2 + 4 = 10$ unit cubes.