This MCQ module is based on: Solids, Nets and Shortest Paths
TOPIC 13 OF 23
Solids, Nets and Shortest Paths
🎓 Class 8
Mathematics
CBSE
Theory
Ch 4 — Playing with Shapes
⏱ ~35 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Solids, Nets and Shortest Paths
Targeting Class 8 level in General Mathematics, with Basic difficulty.
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4.2 Solids: Cuboids, Prisms and Pyramids
There are multiple types of prisms and pyramids, whose names depend on the shapes of their faces.
Prism
A prism?, in general, has two congruent polygons as opposite faces, with edges connecting the corresponding vertices of these polygons. All the other faces are parallelograms. Based on the shape of the congruent polygons, prisms may be called triangular prism, pentagonal prism, and so on.
Pyramid
A pyramid?, in general, has a polygonal base and a point outside it, and edges connect the point with each of the vertices of the base. Based on the shape of the base, pyramids may be called triangular pyramid, square pyramid, pentagonal pyramid, hexagonal pyramids, and so on. A triangular pyramid is also known as a tetrahedron?.
Common prisms and pyramids.
If the congruent polygons of a prism have 10 sides, how many faces, edges and vertices does the prism have?
Prism with \(n\)-sided base has:
Prism with \(n\)-sided base has:
- Faces: \(n + 2 = 12\)
- Edges: \(3n = 30\)
- Vertices: \(2n = 20\)
If the base of a pyramid has 10 sides, how many faces, edges and vertices does the pyramid have?
Pyramid with \(n\)-sided base: Faces = \(n+1 = 11\), Edges = \(2n = 20\), Vertices = \(n+1 = 11\).
Pyramid with \(n\)-sided base: Faces = \(n+1 = 11\), Edges = \(2n = 20\), Vertices = \(n+1 = 11\).
Nets of a Solid
Now we will see how to make different solids using a foldable flat surface such as paper, cardboard, etc. The basic idea is to create a shape on a flat surface that can be folded into solid. Such a shape is called a net?. In other words, a net is obtained by 'unfolding' a solid onto a plane.
A cross-shaped net of a cube (one of 11 possible cube nets).
Net of a sphere? A sphere does not have a net. A sphere has a smooth curved surface that cannot be flattened to a plane without wrinkling, gaps or overlaps (the same mathematical reason world maps always distort).
4.3 Shortest Paths on a Cube
Let us consider an interesting problem related to the discussion so far. We know that on a plane, the shortest path between two points is the straight line between them. Now, what is the shortest path between two points on the surface of a cuboid, if we are allowed to travel only along its surface?
Let us imagine a hungry ant living on the surface of a cuboid. To its good fortune, there is a laddu on the top face. The ant starts on the centre of one side face. What is the shortest path on the surface of the cuboid?
Unfolding the cube turns the curved-surface path into a flat straight line — the shortest route.
If we think that a certain path is the shortest, how can we be sure that it truly is, among all the infinite possibilities? The trick is to unfold the surface onto a plane. On the flat plane, the shortest path is a straight line. So, the route that corresponds to a straight line, when re-folded, is the shortest path on the cube.
Activity: Find the Ant's Shortest Path
Materials: Cardboard cube (5 cm side), string, ruler.
- Mark a point on the middle of one side face (ant) and a point on the middle of the top face (laddu).
- Stretch the string between them along the surface of the cube — explore different routes.
- Now 'unfold' by opening the cube to a net — observe the two points in the flat plane.
- Draw the straight line between them and measure. Compare with the curled paths.
Shortest Path on a Cuboid
Similarly, consider a rectangular cuboid 5 cm × 4 cm × 3 cm. The ant is at one corner of the 5 × 4 face; the laddu is at the opposite corner of the opposite face. Unfold the cuboid into a 5 cm × 7 cm rectangle — the straight diagonal then gives the shortest surface path.
By Pythagoras: shortest path = \(\sqrt{5^2 + 7^2} = \sqrt{74} \approx 8.6\) cm.
Relation Between Faces, Edges and Vertices
| Solid | Faces \(F\) | Edges \(E\) | Vertices \(V\) | \(F+V-E\) |
|---|---|---|---|---|
| Cube | 6 | 12 | 8 | 2 |
| Triangular Prism | 5 | 9 | 6 | 2 |
| Pentagonal Prism | 7 | 15 | 10 | 2 |
| Tetrahedron | 4 | 6 | 4 | 2 |
| Square Pyramid | 5 | 8 | 5 | 2 |
Euler's Formula
For any convex polyhedron: \(F + V - E = 2\).
Competency-Based Questions
Scenario: A packaging engineer designs boxes (cuboids) and pyramids. A cardboard cuboid has dimensions 6 cm × 4 cm × 3 cm. An ant placed at one bottom corner wants to reach the diagonally opposite top corner along the surface.
Q1. Find shortest surface path length (cuboid 6×4×3).
L3Unfold & take best pairing. Lengths: \(\sqrt{6^2 + 7^2} = \sqrt{85}\approx 9.22\), \(\sqrt{4^2 + 9^2}=\sqrt{97}\approx 9.85\), \(\sqrt{3^2+10^2}=\sqrt{109}\approx 10.44\). Minimum ≈ 9.22 cm.
Q2. Analyse: verify Euler's formula for a pentagonal pyramid.
L4F = 6 (5 triangles + 1 base), V = 6, E = 10. F + V − E = 6 + 6 − 10 = 2 ✓.
Q3. Evaluate: a student claims any cube net has exactly 6 squares — so "every 6-square shape is a cube net". Correct?
L5No. Only 11 distinct 6-square patterns fold into a cube. Shapes like 3 × 2 rectangle of squares do NOT fold into a cube. The claim is incorrect.
Q4. Create a net for a square pyramid with base 4 cm and slant height 6 cm. Label all dimensions.
L6A central 4 × 4 square base + 4 isosceles triangles (base 4 cm, slant 6 cm) attached to each side. Total: 5 faces in a plus-like layout.
Assertion–Reason Questions
A: A sphere has no net.
R: A sphere's surface cannot be flattened onto a plane without distortion.
R: A sphere's surface cannot be flattened onto a plane without distortion.
(a) R is the standard Gaussian-curvature justification.
A: For any convex polyhedron, \(F + V - E = 2\).
R: This is Euler's formula for 3-D solids.
R: This is Euler's formula for 3-D solids.
(a) Euler's formula is the direct reason.
Frequently Asked Questions
What is a net of a solid?
A net is a 2D pattern that folds to form a 3D solid. A cube has several possible nets made of six squares arranged so they fold into a cube. NCERT Class 8 Ganita Prakash Part 2 Chapter 4 explores nets of various solids.
How many different nets does a cube have?
A cube has 11 distinct nets. Each is a different arrangement of 6 squares that folds into a cube. Verifying all 11 is a classic NCERT Class 8 Ganita Prakash Part 2 Chapter 4 activity.
How do you find the shortest path on a cube's surface?
Unfold the cube along the faces the path crosses. The shortest surface path becomes a straight line on the unfolded net. Apply Pythagoras' theorem to measure it. NCERT Class 8 Chapter 4 uses this technique.
What solids can be unfolded into nets?
Any polyhedron (cube, cuboid, pyramid, prism, tetrahedron) and also cylinders and cones can be unfolded into flat nets. Spheres cannot be unfolded exactly. NCERT Class 8 Ganita Prakash Part 2 Chapter 4 treats polyhedra.
Why unfold a solid?
Unfolding turns 3D problems (surface area, shortest path) into simpler 2D geometry. It reveals structure and is a powerful problem-solving tool. NCERT Class 8 Chapter 4 makes unfolding a key skill.
What is the shortest path on a cube from corner to opposite corner (on surface)?
For a unit cube, unfold two adjacent faces into an L shape. The straight-line distance is sqrt(1^2 + 2^2) = sqrt(5). This beats routes along edges. NCERT Class 8 Ganita Prakash Part 2 Chapter 4 works through this example.
Frequently Asked Questions — Chapter 4
What is Solids, Nets and Shortest Paths in NCERT Class 8 Mathematics?
Solids, Nets and Shortest Paths is a key concept covered in NCERT Class 8 Mathematics, Chapter 4: Chapter 4. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.
How do I solve problems on Solids, Nets and Shortest Paths step by step?
To solve problems on Solids, Nets and Shortest Paths, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 8 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.
What are the most important formulas for Chapter 4: Chapter 4?
The essential formulas of Chapter 4 (Chapter 4) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.
Is Solids, Nets and Shortest Paths important for the Class 8 board exam?
Solids, Nets and Shortest Paths is part of the NCERT Class 8 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.
What mistakes should students avoid in Solids, Nets and Shortest Paths?
Common mistakes in Solids, Nets and Shortest Paths include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.
Where can I find more NCERT practice questions on Solids, Nets and Shortest Paths?
End-of-chapter NCERT exercises for Solids, Nets and Shortest Paths cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 4, and solve at least one previous-year board paper to consolidate your understanding.
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