Three-Dimensional Shapes
Look around your room. A book is shaped like a cuboid. A ball is a sphere. A cold drink can is a cylinder. An ice cream cone is a cone. A dice is a cube. All these objects have length, width, and height — they are three-dimensional (3D) shapes.
Flat shapes like squares, circles, and triangles are called two-dimensional (2D) shapes because they have only length and width. But the objects we touch and hold in real life have a third dimension — depth or height. These are 3D shapes.
3D shapes have special parts: faces (flat surfaces), edges (lines where faces meet), and vertices (corners where edges meet).
In this chapter, you will learn about common 3D shapes, how to count their faces, edges, and vertices, and how they are different from 2D shapes.
What is Three-Dimensional Shapes - Grade 6 Maths (Understanding Elementary Shapes)?
Definition: A three-dimensional shape (or 3D shape or solid shape) is a shape that has length, width, and height. It occupies space and has volume.
Key terms:
- Face: A flat or curved surface of a 3D shape. Example: A cube has 6 faces.
- Edge: The line where two faces meet. Example: A cube has 12 edges.
- Vertex (plural: vertices): The point where two or more edges meet (a corner). Example: A cube has 8 vertices.
2D vs 3D shapes:
- 2D shapes are flat — they have only length and width (like a square drawn on paper).
- 3D shapes are solid — they have length, width, and height (like a box you can hold).
- 2D shapes have area but no volume. 3D shapes have both area (surface area) and volume.
Three-Dimensional Shapes Formula
Common 3D shapes and their properties:
| Shape | Faces | Edges | Vertices |
| Cube | 6 | 12 | 8 |
| Cuboid | 6 | 12 | 8 |
| Cylinder | 3 (2 flat + 1 curved) | 2 | 0 |
| Cone | 2 (1 flat + 1 curved) | 1 | 1 |
| Sphere | 1 (curved) | 0 | 0 |
| Triangular prism | 5 | 9 | 6 |
| Triangular pyramid (Tetrahedron) | 4 | 6 | 4 |
| Square pyramid | 5 | 8 | 5 |
Euler's Formula (for polyhedra):
F + V - E = 2
Where F = Faces, V = Vertices, E = Edges.
Derivation and Proof
Understanding each 3D shape:
Cube:
- All 6 faces are equal squares.
- All 12 edges are equal in length.
- Example: A dice, a Rubik's cube, a sugar cube.
Cuboid:
- Has 6 rectangular faces (opposite faces are equal).
- Has 12 edges (opposite edges are equal).
- Example: A brick, a book, a matchbox, a mobile phone box.
Cylinder:
- Has 2 flat circular faces (top and bottom) and 1 curved surface.
- Has 2 circular edges and 0 vertices.
- Example: A pipe, a can, a glass, a pillar.
Cone:
- Has 1 flat circular face (base) and 1 curved surface.
- Has 1 circular edge and 1 vertex (the pointed tip, called the apex).
- Example: An ice cream cone, a party hat, a traffic cone.
Sphere:
- Has 1 curved surface (no flat face, no edge, no vertex).
- Every point on the surface is the same distance from the centre.
- Example: A ball, a globe, a marble, the Earth.
Verifying Euler's Formula for a cube:
- F = 6, V = 8, E = 12
- F + V - E = 6 + 8 - 12 = 2 (verified)
Types and Properties
Classification of 3D shapes:
Type 1: Prisms
- Have two identical parallel bases connected by rectangular faces.
- Named by their base shape: triangular prism, rectangular prism (cuboid), pentagonal prism.
- A cube is a special rectangular prism where all faces are squares.
Type 2: Pyramids
- Have one base and triangular faces that meet at a single point (apex).
- Named by their base shape: triangular pyramid, square pyramid, pentagonal pyramid.
- The Egyptian pyramids are square pyramids.
Type 3: Curved Solids
- Have at least one curved surface.
- Cylinder, cone, and sphere are curved solids.
- They do not have all flat faces.
Type 4: Polyhedra vs Non-Polyhedra
- Polyhedra: 3D shapes with all flat faces (cube, cuboid, prisms, pyramids).
- Non-Polyhedra: 3D shapes with curved surfaces (cylinder, cone, sphere).
- Euler's formula (F + V - E = 2) applies only to polyhedra.
Solved Examples
Example 1: Example 1: Identifying 3D Shapes
Problem: Name the 3D shape of each object: (a) a cricket ball (b) a brick (c) a dice (d) a birthday cap (e) a water pipe.
Solution:
- (a) Cricket ball — Sphere
- (b) Brick — Cuboid
- (c) Dice — Cube
- (d) Birthday cap — Cone
- (e) Water pipe — Cylinder
Example 2: Example 2: Counting Faces, Edges, Vertices of a Cuboid
Problem: How many faces, edges, and vertices does a cuboid have?
Solution:
- Faces: 6 (top, bottom, front, back, left, right — all rectangles)
- Edges: 12 (4 on top, 4 on bottom, 4 vertical ones)
- Vertices: 8 (4 on top, 4 on bottom)
Check Euler's Formula: 6 + 8 - 12 = 2 (correct).
Example 3: Example 3: Verifying Euler's Formula for a Triangular Prism
Problem: A triangular prism has 5 faces, 9 edges, and 6 vertices. Verify Euler's formula.
Solution:
- F + V - E = 5 + 6 - 9 = 2
Euler's formula is verified.
Example 4: Example 4: Difference Between 2D and 3D
Problem: What is the difference between a circle and a sphere?
Solution:
- A circle is a 2D shape — it is flat, has no thickness, and can be drawn on paper. It has area but no volume.
- A sphere is a 3D shape — it is a solid ball with length, width, and height. It has both surface area and volume.
- A circle is the 2D cross-section of a sphere.
Example 5: Example 5: Square Pyramid
Problem: Count the faces, edges, and vertices of a square pyramid.
Solution:
- Faces: 1 square base + 4 triangular faces = 5 faces
- Edges: 4 edges on the base + 4 edges from base corners to apex = 8 edges
- Vertices: 4 corners of the base + 1 apex = 5 vertices
Check: F + V - E = 5 + 5 - 8 = 2 (correct).
Example 6: Example 6: Identifying the Shape from Properties
Problem: I have 6 faces, all of which are squares. All my edges are equal. What shape am I?
Solution:
- 6 faces that are all squares → Cube
- A cuboid has 6 faces too, but they are rectangles (not all squares).
Answer: The shape is a cube.
Example 7: Example 7: Curved Surfaces
Problem: Which of these shapes have curved surfaces: cube, cylinder, sphere, cuboid, cone?
Solution:
- Cube — all flat faces (no curved surface)
- Cylinder — has 1 curved surface
- Sphere — has 1 curved surface (entire surface is curved)
- Cuboid — all flat faces (no curved surface)
- Cone — has 1 curved surface
Answer: Cylinder, sphere, and cone have curved surfaces.
Example 8: Example 8: Using Euler's Formula to Find Missing Value
Problem: A 3D shape has 7 faces and 10 vertices. How many edges does it have?
Solution:
- Using Euler's Formula: F + V - E = 2
- 7 + 10 - E = 2
- 17 - E = 2
- E = 17 - 2 = 15
Answer: The shape has 15 edges.
Example 9: Example 9: Real-Life 3D Shapes
Problem: List 3 real-life objects for each: (a) cylinder (b) cuboid (c) sphere.
Solution:
- (a) Cylinder: water bottle, chalk piece, candle
- (b) Cuboid: textbook, eraser, tiffin box
- (c) Sphere: football, orange, marble
Example 10: Example 10: Prism vs Pyramid
Problem: What is the difference between a triangular prism and a triangular pyramid?
Solution:
- A triangular prism has 2 triangular bases connected by 3 rectangular faces. It has 5 faces, 9 edges, 6 vertices.
- A triangular pyramid (tetrahedron) has 1 triangular base and 3 triangular faces meeting at a single point (apex). It has 4 faces, 6 edges, 4 vertices.
- Key difference: A prism has 2 parallel bases; a pyramid has 1 base and an apex.
Real-World Applications
3D shapes in daily life:
- Packaging: Most boxes are cuboids. Cans are cylinders. Understanding shapes helps in designing packages that use less material.
- Architecture: Buildings use cuboid shapes. Domes are hemispheres. Rooftops can be pyramids or prisms.
- Sports: A cricket ball and football are spheres. A shuttlecock has a cone shape.
- Food: A pizza box is a cuboid. An ice cream cone is a cone. A watermelon is roughly a sphere.
- Science: Crystals form cube and prism shapes. Planets are spheres. Understanding 3D shapes is essential in chemistry and physics.
- Art: Sculptors and designers work with 3D shapes. Computer graphics use 3D modelling of these basic shapes.
Key Points to Remember
- 3D shapes have length, width, and height. They occupy space.
- Faces are flat or curved surfaces. Edges are where faces meet. Vertices are corners.
- A cube has 6 square faces, 12 edges, and 8 vertices.
- A cuboid has 6 rectangular faces, 12 edges, and 8 vertices.
- A cylinder has 2 flat circular faces, 1 curved surface, and 0 vertices.
- A cone has 1 flat circular face, 1 curved surface, and 1 vertex (apex).
- A sphere has 1 curved surface, 0 edges, and 0 vertices.
- Euler's Formula: F + V - E = 2 (applies to polyhedra only).
- Prisms have 2 parallel bases; pyramids have 1 base and an apex.
- 2D shapes are flat (area only); 3D shapes are solid (surface area + volume).
Practice Problems
- Name the 3D shape: (a) a tent (b) a globe (c) a pencil without the tip (d) an egg (approximately).
- How many faces, edges, and vertices does a pentagonal prism have?
- A shape has 6 faces, 12 edges, and 8 vertices. Verify Euler's formula and name the shape.
- List all 3D shapes that have at least one curved surface.
- A hexagonal prism has 8 faces. How many edges and vertices does it have? (Use Euler's formula if needed.)
- What 3D shape has exactly 1 vertex?
- Sort these into 2D and 3D: circle, cube, triangle, sphere, rectangle, cylinder.
- A toy is made by placing a cone on top of a cylinder. How many faces, edges, and vertices does the combined shape have?
Frequently Asked Questions
Q1. What is a 3D shape?
A 3D (three-dimensional) shape has length, width, and height. It occupies space and has volume. Examples include cubes, spheres, cylinders, and cones. Objects you can pick up and hold are 3D shapes.
Q2. What is the difference between 2D and 3D shapes?
2D shapes are flat — they have only length and width (like a circle or rectangle drawn on paper). 3D shapes have length, width, AND height — they are solid objects (like a ball or a box).
Q3. What are faces, edges, and vertices?
A face is a flat or curved surface of a 3D shape. An edge is the line where two faces meet. A vertex is the point where edges meet (a corner). A cube has 6 faces, 12 edges, and 8 vertices.
Q4. What is Euler's formula?
Euler's formula states that for any polyhedron (3D shape with all flat faces): Faces + Vertices - Edges = 2 (F + V - E = 2). For example, a cube: 6 + 8 - 12 = 2.
Q5. Does Euler's formula work for a cylinder?
No. Euler's formula applies only to polyhedra (shapes with all flat faces). A cylinder has a curved surface, so it is not a polyhedron. Euler's formula does not apply to cylinders, cones, or spheres.
Q6. What is the difference between a cube and a cuboid?
A cube has 6 faces that are ALL equal squares. A cuboid has 6 faces that are rectangles (opposite faces are equal, but all faces need not be equal). A cube is a special type of cuboid.
Q7. How many vertices does a sphere have?
A sphere has 0 vertices. It has no corners, no edges, and no flat faces. Its entire surface is one smooth curved surface.
Q8. What is the difference between a prism and a pyramid?
A prism has 2 identical parallel bases connected by rectangular faces. A pyramid has 1 base and triangular faces that meet at a single point (apex). A prism looks like a solid bar; a pyramid looks like a pointed structure.
Q9. Is a cone a polyhedron?
No. A cone has a curved surface, so it is not a polyhedron. Polyhedra have only flat faces. The cone is classified as a non-polyhedron or a curved solid.
Q10. What is a net of a 3D shape?
A net is a flat pattern that can be folded to form a 3D shape. For example, the net of a cube is 6 connected squares arranged in a cross-like pattern. When you fold the net, you get the cube.
