How to Find Faces, Edges, and Vertices of 3D Shapes?
The concept of faces, edges and vertices plays a key role in mathematics and is widely applicable in geometry, exams, and real-world problem solving. Understanding these basic properties helps students confidently identify, count, and solve questions on 3D shapes in both classrooms and competitive exams.
What Is Faces, Edges and Vertices?
Faces, edges and vertices are the three main features of 3D (three-dimensional) shapes in maths. A face is a flat or curved surface. An edge is the line where two faces meet. A vertex (plural: vertices) is a sharp corner where two or more edges meet. You’ll find this concept used in geometry, solid shapes, and topics like surface area and volume.
Key Formula for Faces, Edges and Vertices
Here’s the standard formula for polyhedra, known as Euler’s Formula: \( F + V - E = 2 \), where F = number of faces, V = number of vertices, and E = number of edges.
Types of 3D Shapes & Their Faces, Edges, and Vertices
Several common 3D shapes appear in maths and everyday life. Each has its fixed count of faces, edges, and vertices. Recognizing these helps in quick recall for exams and practical usage.
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Cuboid | 6 | 12 | 8 |
| Triangular Prism | 5 | 9 | 6 |
| Square Pyramid | 5 | 8 | 5 |
| Cylinder | 3 | 2 | 0 |
| Cone | 2 | 1 | 1 |
| Sphere | 1 | 0 | 0 |
Step-by-Step Illustration
Let’s see how to identify faces, edges and vertices using a cube as an example:
- Identify the faces: Count all the flat surfaces on the cube. A cube has 6 faces (each one a square).
- Count the edges: Look for all the lines where two faces meet. A cube has 12 edges.
- Locate the vertices: Find the corners where edges meet. A cube has 8 vertices.
Verification with Euler’s Formula:
- Identify values (Cube): F = 6, V = 8, E = 12.
- Use the formula: \( F + V - E = 2 \) → 6 + 8 - 12 = 2Formula confirmed!
Common Confusions: Curved Surfaces, Cylinders & Cones
- Cylinders have 2 flat faces (circles) and 1 curved face (the side). They have 2 edges (where the side meets the circles) but 0 vertices.
- Cones have 1 flat face (base), 1 curved face (side), 1 edge (circle boundary), and 1 vertex (the tip).
- Spheres have 1 curved face, no edges, and no vertices.
- Note: Euler's formula does not work for shapes with curved faces (like a sphere, cylinder, or cone)—only for polyhedra (flat surfaces and straight edges).
Speed Trick or Vedic Shortcut
Here’s a simple shortcut to remember the properties of some common 3D shapes:
- Cube/Cuboid: Always 6 faces, 12 edges, and 8 vertices.
- Cylinder: No corners, always 2 edges, and 0 vertices.
- Pyramid: Number of triangle faces = number of vertices at the base; add 1 face for the base.
School students preparing for CBSE or Olympiads can use quick tables or “real-life shape spotting” (like comparing a can, dice, or ice-cream cone) to remember these counts much faster. Vedantu tutors often use flashcards and online quizzes in live classes to reinforce these shortcuts.
Try These Yourself
- How many faces, edges, and vertices does a triangular prism have?
- Does a cone have any edges?
- List the faces, edges, and vertices in a cuboid found in your house (like a brick or box).
- Identify a real object that looks like a cylinder and write its properties.
Frequent Errors and Misunderstandings
- Counting curved surfaces as “edges” for cylinders and cones—remember, only where flat and curved faces meet can be called an edge.
- Forgetting that spheres have no edges or vertices at all (the surface is fully curved).
- Trying to use Euler’s formula on non-polyhedral shapes (cylinders, cones, spheres)—won’t work!
- Mixing up the difference between “face” (surface) and “side” (may be used for 2D shapes).
Relation to Other Concepts
The idea of faces, edges and vertices closely connects with surface area, volume, solid nets, and even advanced topics like topology and computer graphics. Mastering these basics makes it easier to handle surface-area and volume calculations and helps in solving geometry questions in higher classes.
Faces, Edges and Vertices in Real Life
- Dice: Cube—6 faces, 12 edges, 8 vertices
- Ice-cream cone: Cone—2 faces (1 flat, 1 curved), 1 edge, 1 vertex
- Soda can: Cylinder—3 faces (2 flat, 1 curved), 2 edges, 0 vertices
- Football: Sphere—1 face, 0 edges, 0 vertices
- Box: Cuboid—6 faces, 12 edges, 8 vertices
Quick Reference Table: Major 3D Shapes
| Solid Name | Vertices | Faces | Edges |
|---|---|---|---|
| Cube | 8 | 6 | 12 |
| Cuboid | 8 | 6 | 12 |
| Triangular Prism | 6 | 5 | 9 |
| Rectangular Pyramid | 5 | 5 | 8 |
| Cylinder | 0 | 3 | 2 |
| Cone | 1 | 2 | 1 |
| Sphere | 0 | 1 | 0 |
Classroom Tip
A fun way to remember faces, edges and vertices is:“Faces are flat or curved; edges are where faces meet; vertices are pointy corners.” Build models with clay, straws, and paper. Vedantu’s live sessions use interactive nets and real-life objects for practice and revision. Download summary charts and worksheets for extra practice.
Exam Practice and Worksheet
Practice is the key! Use CBSE and ICSE exam sample questions, solve MCQs based on faces, edges and vertices, and try fill-in-the-blanks exercises.
Sample Problem:
1. A square pyramid has how many faces, edges, and vertices?2. Count the vertices of a hexagonal prism.
3. Quick: Does a sphere have any edges?
We explored faces, edges and vertices—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept.
Suggested Internal Links
- Euler's Formula – Explains how faces, edges, and vertices are connected
- Cube – Full cube breakdown and its properties
- Solid Geometry – Learn about 3D shapes and broader geometry concepts
FAQs on Faces, Edges and Vertices in Maths – Definitions, Easy Tricks & Examples
1. What are faces, edges, and vertices in 3D shapes?
In geometry, faces are the flat or curved surfaces of a three-dimensional shape. Edges are the line segments where two faces meet. Vertices (singular: vertex) are the points where two or more edges intersect—essentially, the corners of the shape.
2. How do you identify the faces, edges, and vertices of a cube?
A cube has six square faces, twelve edges (line segments where faces meet), and eight vertices (corners where edges meet).
3. How many faces, edges, and vertices does a triangular prism have?
A triangular prism has five faces (two triangles and three rectangles), nine edges, and six vertices.
4. What is Euler's formula, and how does it relate to faces, edges, and vertices?
Euler's formula is a fundamental relationship in geometry that states: V - E + F = 2, where 'V' represents the number of vertices, 'E' represents the number of edges, and 'F' represents the number of faces. This formula applies to convex polyhedra (3D shapes with flat faces and straight edges).
5. How many faces, edges, and vertices does a square pyramid have?
A square pyramid has five faces (one square base and four triangular faces), eight edges, and five vertices.
6. Does Euler's formula work for all 3D shapes? Explain.
No, Euler's formula (V - E + F = 2) does not apply to all 3D shapes. It only works for convex polyhedra. Shapes with curved surfaces, like spheres or cylinders, do not satisfy this formula.
7. What are some real-world examples of objects with faces, edges, and vertices?
Many everyday objects illustrate these concepts: a dice (cube), a shoebox (cuboid), a pyramid (Egyptian pyramids), a tent (triangular prism), and even a crystal can demonstrate various combinations of faces, edges, and vertices.
8. How do you count faces, edges, and vertices on complex 3D shapes?
For complex shapes, systematically count each feature. Start with one face and trace its edges to adjacent faces, carefully counting each unique vertex. A methodical approach prevents double-counting and ensures accuracy. Using Euler's formula as a check can also help if applicable.
9. What are the faces, edges, and vertices of a cone?
A cone is a special case. It typically has one circular base (considered a face), one curved lateral surface (also a face), one apex (vertex), and one edge which is the circumference of the base.
10. What are the faces, edges, and vertices of a cylinder?
A cylinder usually has two circular bases (faces), one curved lateral surface (face), no vertices, and two edges (the circumferences of the bases).
11. What are the faces, edges, and vertices of a sphere?
A sphere has one curved surface (face), no edges, and no vertices.
12. How are faces, edges, and vertices used in computer graphics?
In computer graphics, faces, edges, and vertices are fundamental elements for creating and manipulating 3D models. They define the shape's geometry, allowing for rendering, animation, and interaction.
