10.2 2D & 3D Shape Properties

Geometry and spatial reasoning on the ISEB Common Pre-Test require a synthesis of deductive logic and computational precision. The mathematics syllabus expects candidates to analyze two-dimensional polygons and three-dimensional polyhedra with rigorous mathematical vocabulary, estimate and measure angles, use straight-line and whole-turn totals, and identify shapes from their properties, and manipulate geometric figures within the first quadrant of the Cartesian coordinate plane.

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1. Two-Dimensional Shape Properties

Triangle Taxonomy

Triangles are three-sided polygons classified both by their side lengths and by their interior angles. The interior angles of any triangle on a flat plane sum to $180^\circ$.

Quadrilaterals: Taxonomy & Diagonal Properties

A quadrilateral is a four-sided polygon whose interior angles always sum to $360^\circ$. The table describes the typical non-special cases. A square is also a rectangle and a rhombus, and both are parallelograms. For rows other than square, the stated counts exclude more symmetric special cases. We use the common UK definition of trapezium with exactly one pair of parallel sides:

Regular vs. Irregular Polygons

A polygon is regular if and only if it satisfies two conditions simultaneously:

1. It is equilateral (all sides have identical length).
2. It is equiangular (all interior angles have identical magnitude).

Optional extension: polygon angle formulae

For any convex polygon with $n$ sides:

$$\begin{aligned} \text{Sum of Interior Angles} &= (n - 2) \times 180^\circ \\ \text{Regular Polygon Interior Angle} &= \frac{(n - 2) \times 180^\circ}{n} \\ \text{Sum of Exterior Angles} &= 360^\circ \quad \text{(for a complete turn around a convex polygon)} \\ \text{Regular Polygon Exterior Angle} &= \frac{360^\circ}{n} \end{aligned}$$

For the convex polygons considered here, an interior angle and its adjacent exterior angle form a straight line:

$$\text{Interior Angle} + \text{Exterior Angle} = 180^\circ$$

Polygon Benchmark Properties

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2. Three-Dimensional Shape Properties

A polyhedron (plural: polyhedra) is a three-dimensional solid bounded entirely by flat polygonal faces that meet at straight edges, which intersect at sharp points called vertices.

Prisms vs. Pyramids

• Right prism: A solid with two congruent, parallel polygonal bases connected by rectangular lateral faces. A general oblique prism can instead have parallelogram lateral faces. The cross-section along its entire length is uniform.
  • For an $n$-sided base polygon: $\text{Faces} = n + 2$, $\text{Vertices} = 2n$, $\text{Edges} = 3n$.
• Pyramid: A solid with a single polygonal base connected by triangular lateral faces tapering to a single point called the apex.
  • For an $n$-sided base polygon: $\text{Faces} = n + 1$, $\text{Vertices} = n + 1$, $\text{Edges} = 2n$.

Optional extension: Euler's polyhedron formula

For any convex polyhedron, Swiss mathematician Leonhard Euler proved the universal relationship:

$$\text{Faces} + \text{Vertices} - \text{Edges} = 2 \quad \text{or} \quad F + V = E + 2$$

┌────────────────────────────────────────────────────────────────────────┐
│                     Euler's Polyhedron Check                           │
├───────────────────────────┬─────────┬────────────┬─────────┬───────────┤
│ Polyhedron                │ Faces F │ Vertices V │ Edges E │ F + V - E │
├───────────────────────────┼─────────┼────────────┼─────────┼───────────┤
│ Cube / Cuboid             │    6    │     8      │   12    │ 6+8-12=2  │
│ Triangular Prism          │    5    │     6      │    9    │ 5+6-9=2   │
│ Square-Based Pyramid      │    5    │     5      │    8    │ 5+5-8=2   │
│ Tetrahedron (Tri. Pyramid)│    4    │     4      │    6    │ 4+4-6=2   │
│ Pentagonal Prism          │    7    │    10      │   15    │ 7+10-15=2 │
│ Hexagonal Prism           │    8    │    12      │   18    │ 8+12-18=2 │
│ Octahedron                │    8    │     6      │   12    │ 8+6-12=2  │
└───────────────────────────┴─────────┴────────────┴─────────┴───────────┘

Curved Solids (Non-Polyhedra)

Non-polyhedra possess curved surfaces and do not strictly follow Euler's polyhedral formula:

• Cylinder: 2 flat circular faces, 1 curved surface, 2 circular edges, 0 vertices.
• Cone: 1 flat circular face, 1 curved surface, 1 circular edge, 1 apex (vertex).
• Sphere: 0 flat faces, 1 completely curved surface, 0 edges, 0 vertices.
• Hemisphere: 1 flat circular face, 1 curved surface, 1 circular edge, 0 vertices.

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Solid deduction extension

Worked Example 3: Euler's Polyhedron Deduction

Problem: A geometric solid is formed by joining two identical square-based pyramids base-to-base (creating an octahedron; the pyramids need suitable equal edge lengths for it to be regular). How many faces, edges, and vertices does this solid have, and does it satisfy Euler's formula?

• Step 1 (Count faces):
  • Each square-based pyramid has 4 triangular faces. When joined base-to-base, the two square bases become internal and are no longer external faces.
  • Total external faces $= 4 + 4 = 8$ triangular faces ($F = 8$).
• Step 2 (Count vertices):
  • There is 1 top apex, 1 bottom apex, and 4 vertices around the middle equatorial square.
  • Total vertices $= 1 + 1 + 4 = 6$ vertices ($V = 6$).
• Step 3 (Count edges):
  • 4 edges lead from top apex to the square; 4 edges lead from bottom apex to the square; 4 edges form the equatorial square.
  • Total edges $= 4 + 4 + 4 = 12$ edges ($E = 12$).
• Step 4 (Test Euler's formula): $$F + V - E = 8 + 6 - 12 = 14 - 12 = 2$$
• Answer: Faces $= 8$, Vertices $= 6$, Edges $= 12$; Euler's formula is satisfied ($8 + 6 - 12 = 2$).