Exercise 13.2 — Euler's Theorem

What is Exercise 13.2 About?

Exercise 13.2 of Class 8 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) is part of Chapter 13 — Visualizing 3-D Shapes in 2-D. This exercise dives deep into the structure of three-dimensional solid objects — understanding polyhedrons and non-polyhedrons, identifying faces, edges, and vertices, and applying the landmark Euler's Formula (F + V = E + 2) to verify properties of various solids.

The exercise covers classification of solids, types of prisms and pyramids, Euler's relation verification across eight different polyhedra, reasoning questions, table completion using Euler's formula, and identifying 3-D shapes from their 2-D nets. Mastering this topic is essential for scoring full marks in board exams across CBSE, Telangana, and Andhra Pradesh.

Polyhedron and Non-Polyhedron — Core Definitions

The most fundamental classification in this chapter divides all 3-D objects into two groups based on whether their faces are flat or curved.

Here is a quick visual guide to recognizing which common shapes belong to which group:

Faces, Edges, and Vertices — Understanding the Building Blocks

Every polyhedron is described by three fundamental measurements: Faces (F) — the flat polygonal surfaces; Edges (E) — the straight line segments where two faces meet; and Vertices (V) — the corner points where three or more edges meet.

• Face (F) — A flat surface of the polyhedron. A cube has 6 faces, all of which are squares.
• Edge (E) — A line segment formed where two faces join. A cube has 12 edges.
• Vertex (V) — A corner point where edges meet. A cube has 8 vertices.

The famous Swiss mathematician Leonard Euler discovered a beautiful relationship that connects these three numbers for any convex polyhedron:

The table below summarises F, V, E values for common polyhedrons and confirms Euler's relation in every case:

Regular Polyhedrons, Prisms, and Pyramids

Polyhedrons can be further classified based on whether their faces are all congruent, and by their structural type (prisms or pyramids).

Prisms

A prism is a solid that has two parallel, congruent polygonal bases connected by rectangular (or parallelogram) lateral faces. The name of the prism comes from the shape of its base.

Pyramids

A pyramid has a polygonal base and triangular lateral faces that all meet at a single point called the apex.

Question 1 — Verify Euler's Relation for 8 Polyhedra

This question presents eight different polyhedra and asks you to count their faces, vertices, and edges, then verify that F + V = E + 2 holds in every case. Each solution below shows the complete step-by-step working.

Questions 2 & 3 — Reasoning About Polyhedra

Question 5 — Complete the Table Using Euler's Formula

This question gives a table with some values of F, V, and E missing. You must rearrange Euler's formula to find the unknown value in each case.

Question 6 — Can a Polyhedron Have 10 Faces, 20 Edges, and 15 Vertices?

Question 7 — Complete the Table (Vertices and Edges)

This question shows three objects and asks you to identify their vertices and edges from the diagram. The answers are:

Question 8 — Name the 3-D Shapes Formed by Each Net

A net is a 2-D flat layout that can be folded along its edges to form a 3-D solid. Recognizing which net gives which shape is a key skill tested in board exams across CBSE, Telangana, and Andhra Pradesh.

Question 9 — Which Nets Make a Cube?

There are 11 possible nets of a cube (using 6 connected squares). In this question, among options (a) through (k), the nets that fold correctly into a cube are:

Question 9(ii) — Fill in the Blanks / Short Answer

Question 9(iii) — Name the 3-D Objects

Four 3-D objects are shown and students must identify them by name. The answers are:

Common Mistakes to Avoid in Exercise 13.2

• Miscounting edges: Students often confuse edges with sides of faces. Remember — an edge is the line where two faces physically meet, not just any line drawn on a face.
• Forgetting the apex vertex: In pyramids, students sometimes count only the base vertices and forget the single apex point at the top. A square pyramid has 4 base + 1 apex = 5 vertices.
• Confusing Euler's formula direction: The formula is F + V = E + 2, not F + E = V + 2. Remember: Faces and Vertices on the left; Edges on the right.
• Applying Euler's formula to non-polyhedrons: Euler's relation applies only to convex polyhedrons. It does not apply to spheres, cylinders, or cones.
• Net recognition errors: Always fold the net mentally. Check that no two panels land on the same face position when folded. If any overlap occurs, the net is invalid.
• Calling a square prism a cube: A square prism only becomes a cube when all faces are equal squares. Otherwise it is just a square prism.

What This Exercise Prepares You For

The concepts in Exercise 13.2 build a foundation that extends well beyond Class 8. Euler's formula reappears in Class 9 and 10 geometry and is also a key concept in competitive mathematics. Understanding faces, edges, and vertices is directly required for Exercise 13.1, where you learn to draw isometric sketches and oblique projections of 3-D solids.

The ability to recognize shapes from their nets is tested repeatedly in Class 9 surface area and volume chapters. For example, when computing the surface area of a prism or pyramid, you essentially "unroll" the net and sum the areas of each face panel — the exact skill developed in this exercise.

For Telangana and Andhra Pradesh board examinations, Exercise 13.2 problems — especially Euler's relation verification and net identification — appear regularly as 2-mark and 4-mark questions. Students who can systematically count faces, edges, and vertices and apply the formula fluently have a clear scoring advantage.