Grade 9 Mathematics
Grade 9 · Term 4Mathematics

Geometry of 3D Objects

We investigate the properties of three-dimensional objects including Platonic solids, prisms, pyramids, cylinders, cones, and spheres. We build and interpret nets (flat representations) of 3D solids.

9.1 Properties of 3D Objects9.2 Nets of 3D Objects
Week 1

9.1 Properties of 3D Objects

  • Identify, describe, and name the five Platonic solids and their properties
  • Describe prisms, pyramids, cylinders, cones, and spheres using faces, vertices, and edges
  • Apply Euler's formula: F + V − E = 2
🌍

Real-World Connection

3D shapes are everywhere: ice cubes are rectangular prisms, ice cream cones are cones, soccer balls approximate spheres, Egyptian pyramids are square pyramids, and the molecular structure of carbon in a diamond is a tetrahedron. Architects, engineers, and designers work in 3D constantly — understanding these shapes is the foundation of spatial reasoning.

Definition

Platonic Solids

The five regular convex polyhedra where all faces are identical regular polygons. Named after Plato who associated them with the classical elements.

Tetrahedron (4), Cube (6), Octahedron (8), Dodecahedron (12), Icosahedron (20 faces)\text{Tetrahedron (4), Cube (6), Octahedron (8), Dodecahedron (12), Icosahedron (20 faces)}

Platonic Solids — Properties

Solid

Faces / Edges / Vertices

Tetrahedron

4 triangular faces

F=4, E=6, V=4

Cube (Hexahedron)

6 square faces

F=6, E=12, V=8

Octahedron

8 triangular faces

F=8, E=12, V=6

Dodecahedron

12 pentagonal faces

F=12, E=30, V=20

Icosahedron

20 triangular faces

F=20, E=30, V=12

Euler's Formula

F+V−E=2F + V - E = 2

F = number of faces, V = vertices, E = edges. Works for all convex polyhedra.

Definition

Prism

A solid with two identical, parallel polygonal bases connected by rectangular faces. Named by the shape of its base: triangular prism, rectangular prism (cuboid), etc.

Faces=n+2,  Vertices=2n,  Edges=3n(n-sided base)\text{Faces} = n+2, \; \text{Vertices} = 2n, \; \text{Edges} = 3n \quad (n\text{-sided base})

Definition

Pyramid

A solid with a polygonal base and triangular faces meeting at a single point (apex). Named by base shape: square pyramid, triangular pyramid (tetrahedron), etc.

Faces=n+1,  Vertices=n+1,  Edges=2n(n-sided base)\text{Faces} = n+1, \; \text{Vertices} = n+1, \; \text{Edges} = 2n \quad (n\text{-sided base})
Worked Examples

Worked Example

Verifying Euler's formula

Problem

Verify Euler's formula for a triangular prism.

Worked Example

Classifying and comparing 3D objects

Problem

Compare a cube and a square-based pyramid: list the faces, vertices, edges, and verify Euler's formula for each.

Worked Example

Identifying Platonic solids

Problem

Which Platonic solid has (a) 20 equilateral triangular faces? (b) 12 pentagonal faces? State the number of vertices and edges for each.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
How many faces, vertices, and edges does a cube have?
2
L1 · Knowledge1 mark
Name the 3D solid with a circular base and a curved surface meeting at a single point.
3
L2 · Routine Procedures4 marks
A square pyramid has a square base. Find the number of faces, vertices, and edges. Verify with Euler's formula.
4
L2 · Routine Procedures3 marks
Describe the difference between a prism and a pyramid with the same base.
5
L3 · Complex Procedures3 marks
A polyhedron has 10 faces and 15 edges. How many vertices does it have? Use Euler's formula.
6
L3 · Complex Procedures4 marks
How many faces, edges, and vertices does a pentagonal prism have? Verify with Euler's formula.
7
L4 · Problem Solving5 marks
Research question: Why are there only 5 Platonic solids? Hint: consider how many regular polygons can meet at a vertex.
8
L4 · Problem Solving4 marks
A soccer ball consists of 12 pentagons and 20 hexagons. Verify that this satisfies Euler's formula.
Week 2

9.2 Nets of 3D Objects

  • Identify and draw nets of prisms, pyramids, cylinders, and cones
  • Build 3D models from nets
  • Determine the surface area from a net
🌍

Real-World Connection

Packaging designers use nets every day. A cereal box (rectangular prism), a Toblerone box (triangular prism), and a party hat (cone) all start as flat sheets that are folded. Understanding nets means you can unfold any box in your mind and calculate exactly how much cardboard is needed — no waste, no shortfall.

Definition

Net

A net is a flat (2D) shape that can be folded to form a 3D solid. Every face of the solid appears as a polygon in the net.

Cube net: 6 squares arranged in a cross or T-shape pattern\text{Cube net: 6 squares arranged in a cross or T-shape pattern}

💡 Tip

When drawing nets: (1) Identify all faces and their shapes. (2) Make sure each face is the correct size. (3) Check that adjacent faces in the net will actually be adjacent in the 3D solid. (4) Verify: count faces in the net equals faces in the solid.

A cube has 11 different nets (different arrangements of 6 squares that all fold into a cube). A rectangular prism has 2 pairs of congruent rectangles for the faces. A cylinder's net consists of a rectangle (the curved surface, width = circumference) and two circles.

Worked Examples

Worked Example

Finding the surface area from a net

Problem

A closed cylinder has radius 3 cm and height 10 cm. Describe its net and find the total surface area.

Worked Example

Drawing and identifying nets

Problem

Sketch the net of a square-based pyramid with base side 6 cm and slant height 5 cm. Calculate the total surface area from the net.

Worked Example

Net of a triangular prism

Problem

A triangular prism has equilateral triangle ends with side 4 cm, and length 9 cm. Describe its net and find the total surface area.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge1 mark
How many faces does the net of a triangular prism have?
2
L1 · Knowledge2 marks
Describe the net of a cone.
3
L2 · Routine Procedures4 marks
A rectangular prism has dimensions 5 cm × 4 cm × 3 cm. Describe its net and calculate the surface area.
4
L2 · Routine Procedures4 marks
A square pyramid has base 6 cm × 6 cm and triangular faces with slant height 5 cm. Find its total surface area from the net.
5
L3 · Complex Procedures4 marks
A gift box is a cube with side 20 cm. How much wrapping paper (in cm²) is needed, allowing 10% extra for overlap?
6
L3 · Complex Procedures4 marks
A cylindrical can (open top, no lid) has radius 4 cm and height 12 cm. Find the area of its net.
7
L4 · Problem Solving5 marks
A cereal box has dimensions 30 cm × 8 cm × 20 cm. The cardboard costs R0.05 per cm². Calculate the cost to manufacture 500 boxes.
8
L4 · Problem Solving5 marks
A cone has slant height ll and base radius rr. Its net consists of a sector and a circle. Show that the curved surface area = πrl\pi r l.

Next

Surface Area & Volume

Geometry of 3D Objects Grade 9 Maths CAPS Notes & Examples | MathSciBuddy