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Three-Dimensional Solids, Nets and Orthographic Views Notes for GATE CS

Three-Dimensional Solids, Nets and Orthographic Views notes for GATE CS: 35 study cards covering concepts, formulas, shortcuts and exam traps, plus solved pra

three dimensional solids nets and orthographic views notes

Chapter Roadmap: Transformation of Shapes

Chapter Roadmap: Transformation of Shapes 1. Cube Nets and Solid Formation Identify valid nets, visualize folding, determine face adjacency. Weightage: Moderate | Foundation for 3D visualization 2. Solid Mensuration and Equal-Volume Comparisons Relate surface area and volume across cubes, cuboids, cylinders. Weightage: Moderate | Formula application and reasoning 3. Three-Dimensional Cutting and Subdivision Minimize cuts, predict piece shapes, understand symmetry in division. Weightage: Moderate | Logical optimization 4. Surface Curvature and Solid Geometry Distinguish smooth versus polyhedral surfaces, identify curvature properties. Weightage: Low-Moderate | Conceptual classification 5. Orthographic Views and 3D Reconstruction Interpret front, top, side views; reconstruct 3D from 2D projections. Weightage: Moderate-High | High-value spatial reasoning By the end of this chapter: You will master the mental manipulation of 3D objects, quickly identify valid solid formations, compare geometric properties efficiently, and reconstruct solids from multiple views with confidence.

What is a Cube Net

What is a Cube Net A cube net is a two-dimensional arrangement of six identical squares, connected along their edges, that can be folded along those edges to form a closed three-dimensional cube. Key Properties Exactly 6 squares (one for each face of the cube) Squares are connected edge-to-edge (not corner-to-corner) When folded, no squares overlap and no gaps remain There are exactly 11 distinct nets for a cube Visual Intuition Imagine cutting along the edges of a cardboard cube and flattening it. The resulting pattern is a net. Different cutting paths produce different nets, but only 11 unique arrangements exist. One of the 11 valid cube nets Why This Matters In exams, you are shown a net and asked which cube it forms, or shown a cube and asked which net could produce it. Understanding the folding process is essential.

The Eleven Distinct Cube Nets

The Eleven Distinct Cube Nets While there are many ways to arrange six squares, only 11 distinct nets can fold into a cube. Distinct means that nets that are rotations or mirror images of each other count as one. Common Categories Cross Pattern (1 net): Four squares in a central row, one square above and below the second square. T-Patterns (3 nets): Three squares in a row, additional squares branching off. L-Patterns and Zigzags (7 nets): Various arrangements with bends and turns. Recognition Strategy Instead of memorizing all 11, learn to validate: Count squares: must be exactly 6 Check connectivity: all squares connected edge-to-edge Test folding mentally: can it close without overlap Cross T-variant

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Question 1

Level 1: Warm-up

In a cube net, four squares are arranged in a straight continuous row. If the squares are numbered 1, 2, 3, and 4 from left to right, which pair of squares will become opposite faces when the net is folded into a cube?

Question 2

Level 1: Warm-up

A cube has 12 edges. To unfold a cube into a single connected 2D net without any overlapping faces, exactly how many of its edges must be cut?

Question 3

Level 1: Warm-up

If a solid cuboid is cut by n_{x} planes parallel to the YZ-plane, n_{y} planes parallel to the XZ-plane, and n_{z} planes parallel to the XY-plane, what is the total number of pieces formed?

Question 4

Level 1: Warm-up

A valid cube net is formed by a specific number of identical squares connected edge-to-edge. How many squares are strictly required to form a valid cube net?

Question 5

Level 1: Warm-up

A flat rectangular paper sheet is rolled and its edges are joined to form a hollow cylindrical tube without any overlap. The area of the original flat sheet is exactly equal to which of the following properties of the newly formed cylinder?

Question 6

Level 1: Warm-up

If two squares in a valid 2D cube net share a common edge, what is their spatial relationship when the net is folded into a 3D cube?

Question 7

Level 1: Warm-up

When a solid metal object is melted and recast into a new shape without any material loss or addition, which geometric property is strictly conserved?

Question 8

Level 1: Warm-up

When a solid metal object is melted and recast into a completely different shape without any material loss or addition, which geometric property is strictly conserved?

Question 9

Level 1: Warm-up

When a single straight cut is made completely through a solid, how does the total surface area of all the resulting pieces compare to the original solid's surface area?

Question 10

Level 1: Warm-up

What is the minimum number of straight cuts required to divide a cube into 4 equal-sized pieces, if each cut must go completely through the cube and be parallel to one of its faces?

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Three-Dimensional Solids, Nets and Orthographic Views Notes for GATE CS

Three-Dimensional Solids, Nets and Orthographic Views notes for GATE CS: 35 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Transformation of Shapes

1. Cube Nets and Solid Formation
Identify valid nets, visualize folding, determine face adjacency.
Weightage: Moderate | Foundation for 3D visualization

2. Solid Mensuration and Equal-Volume Comparisons
Relate surface area and volume across cubes, cuboids, cylinders.
Weightage: Moderate | Formula application and reasoning

3. Three-Dimensional Cutting and Subdivision
Minimize cuts, predict piece shapes, understand symmetry in division.
Weightage: Moderate | Logical optimization

4. Surface Curvature and Solid Geometry
Distinguish smooth versus polyhedral surfaces, identify curvature properties.
Weightage: Low-Moderate | Conceptual classification

5. Orthographic Views and 3D Reconstruction
Interpret front, top, side views; reconstruct 3D from 2D projections.
Weightage: Moderate-High | High-value spatial reasoning

By the end of this chapter: You will master the mental manipulation of 3D objects, quickly identify valid solid formations, compare geometric properties efficiently, and reconstruct solids from multiple views with confidence.

What is a Cube Net

A cube net is a two-dimensional arrangement of six identical squares, connected along their edges, that can be folded along those edges to form a closed three-dimensional cube.

Key Properties

Exactly 6 squares (one for each face of the cube)
Squares are connected edge-to-edge (not corner-to-corner)
When folded, no squares overlap and no gaps remain
There are exactly 11 distinct nets for a cube

Visual Intuition

Imagine cutting along the edges of a cardboard cube and flattening it. The resulting pattern is a net. Different cutting paths produce different nets, but only 11 unique arrangements exist.

Why This Matters

In exams, you are shown a net and asked which cube it forms, or shown a cube and asked which net could produce it. Understanding the folding process is essential.

The Eleven Distinct Cube Nets

While there are many ways to arrange six squares, only 11 distinct nets can fold into a cube. Distinct means that nets that are rotations or mirror images of each other count as one.

Common Categories

Cross Pattern (1 net): Four squares in a central row, one square above and below the second square.
T-Patterns (3 nets): Three squares in a row, additional squares branching off.
L-Patterns and Zigzags (7 nets): Various arrangements with bends and turns.

Recognition Strategy

Instead of memorizing all 11, learn to validate:

Count squares: must be exactly 6
Check connectivity: all squares connected edge-to-edge
Test folding mentally: can it close without overlap

Method: Validating a Cube Net

Use this step-by-step approach to determine if a flat pattern is a valid cube net.

Step 1: Count the Squares

Must have exactly 6 squares
If more or fewer, immediately invalid

Step 2: Check Connectivity

All squares must be connected edge-to-edge
Must form a single connected piece
Corner-only connections are invalid

Step 3: Quick Elimination Rules

No row of 5 or more squares: Maximum 4 in a straight line
No square with 4 neighbors: A square cannot be surrounded on all four sides
No isolated branches: Every square must connect to the main structure

Step 4: Mental Folding Test

Choose a base square (usually central)
Identify adjacent squares that fold up
Determine which squares become opposite faces
Check for overlaps: Two squares cannot occupy the same position
Verify closure: All six faces accounted for, no gaps

    Pattern: 4 squares in a row, with one square above the second and one below the third

    Step 1: 6 squares (Pass)

    Step 2: All connected (Pass)

    Step 3: Max 4 in row, no square with 4 neighbors (Pass)

    Step 4: Fold test yields valid net (Pass)

Three-Dimensional Solids, Nets and Orthographic Views: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Spatial Aptitude MCQ

In a cube net, four squares are arranged in a straight continuous row. If the squares are numbered 1, 2, 3, and 4 from left to right, which pair of squares will become opposite faces when the net is folded into a cube?

A.
1 and 2
B.
1 and 3
C.
2 and 3
D.
1 and 4

Correct Answer:

Step-by-Step Solution

Key idea: This is a <direct_formula> question testing the opposite face rules in straight rows.

Step 1: Recall the separation rule for straight rows. In a continuous straight line of squares, any two squares separated by exactly one square will become opposite faces when folded.

Step 2: Apply the rule to the numbered row (1, 2, 3, 4).

Step 3: Square 1 and Square 3 are separated by Square 2. Therefore, 1 and 3 are opposite.

Step 4: Similarly, Square 2 and Square 4 are separated by Square 3. Therefore, 2 and 4 are opposite.

Step 5: Check the options. Option B correctly identifies the pair 1 and 3.

Answer: B

Question 2 · Spatial Aptitude NAT

A cube has 12 edges. To unfold a cube into a single connected 2D net without any overlapping faces, exactly how many of its edges must be cut?

Correct Answer:

Step-by-Step Solution

Key idea: This is a <spatial_folding> question testing the graph theory concept of unfolding a polyhedron into a spanning tree of faces.

Step 1: Identify the total number of faces in a cube. A cube has 6 faces.

Step 2: Determine the number of connections needed. To keep 6 faces connected in a single piece (a tree structure) without forming any closed loops (which would prevent it from lying flat), you need exactly $6 - 1 = 5$ connecting edges (folds).

Step 3: Identify the total number of edges in a cube. A cube has 12 edges.

Step 4: Calculate the cuts. The edges that are not kept as folds must be cut. Number of cuts = Total edges - Folded edges = $12 - 5 = 7$ .

Answer: 7

Question 3 · Spatial Aptitude MCQ

If a solid cuboid is cut by $n_{x}$ planes parallel to the YZ-plane, $n_{y}$ planes parallel to the XZ-plane, and $n_{z}$ planes parallel to the XY-plane, what is the total number of pieces formed?

A.
$n_{x} + n_{y} + n_{z}$
B.
$(n_{x} + 1) (n_{y} + 1) (n_{z} + 1)$
C.
$n_{x} n_{y} n_{z}$
D.
$(n_{x} + 1) + (n_{y} + 1) + (n_{z} + 1)$

Correct Answer:

Step-by-Step Solution

Key idea: This is a <direct_formula> question testing the principle of orthogonal cuts in 3D subdivision.

Step 1: Understand how cuts work in one dimension. $n_{x}$ cuts parallel to a plane divide the solid into $n_{x} + 1$ pieces along that axis.

Step 2: Extend to three dimensions. Since the cuts are orthogonal (independent along x, y, and z axes), the total number of pieces is the product of the number of divisions along each axis.

Step 3: Apply the formula. Total pieces = $(n_{x} + 1) \times (n_{y} + 1) \times (n_{z} + 1)$ .

Answer: B

Question 4 · Spatial Aptitude MCQ

A valid cube net is formed by a specific number of identical squares connected edge-to-edge. How many squares are strictly required to form a valid cube net?

A.
4
B.
5
C.
6
D.
8

Correct Answer:

Step-by-Step Solution

Key idea: This is a <definition> question testing the fundamental property of a cube net.

Step 1: Recall the definition of a cube. A standard cube is a 3D solid with exactly 6 identical square faces.

Step 2: Understand how a net works. A net is a 2D pattern that folds into a 3D solid without gaps or overlaps.

Step 3: Therefore, to cover all 6 faces of the cube exactly once, the net must consist of exactly 6 squares.

Answer: C

Question 5 · Spatial Aptitude MCQ

A flat rectangular paper sheet is rolled and its edges are joined to form a hollow cylindrical tube without any overlap. The area of the original flat sheet is exactly equal to which of the following properties of the newly formed cylinder?

A.
Curved surface area
B.
Total surface area
C.
Volume
D.
Base area

Correct Answer:

Step-by-Step Solution

Key idea: This is a <definition> question testing the principle of conservation of material during shape transformation.

Step 1: Visualize the transformation. A flat 2D rectangle is bent into a 3D cylindrical tube.

Step 2: Identify what the sheet covers. The sheet forms the outer wrapping of the cylinder, which is the lateral or curved surface. It does not form the top or bottom circular bases.

Step 3: Apply the conservation principle. The 2D area of the sheet is conserved and becomes the 3D Curved Surface Area (CSA) of the cylinder.

Answer: A

Question 6 · Spatial Aptitude MCQ

If two squares in a valid 2D cube net share a common edge, what is their spatial relationship when the net is folded into a 3D cube?

A.
They become opposite faces
B.
They become adjacent faces
C.
They become the same face
D.
They do not touch each other

Correct Answer:

Step-by-Step Solution

Key idea: This is a <spatial_folding> question testing the fundamental adjacency rule of cube nets.

Step 1: Understand the definition of a cube net. A net is formed by squares connected edge-to-edge.

Step 2: Visualize the folding process. When two squares share a common edge in the 2D net, that edge becomes the hinge or fold line.

Step 3: Conclude the relationship. Because they are connected by a fold line, they will naturally form two faces that meet at that edge in the 3D cube, making them adjacent faces.

Answer: B

Question 7 · Spatial Aptitude MCQ

When a solid metal object is melted and recast into a new shape without any material loss or addition, which geometric property is strictly conserved?

A.
Total surface area
B.
Total volume
C.
Total edge length
D.
Curved surface area

Correct Answer:

Step-by-Step Solution

Key idea: This is a <conservation> question testing the fundamental principle of material transformation.

Step 1: Identify the type of transformation. The keywords "melted and recast" indicate a change in state and shape, but the amount of material remains the same.

Step 2: Apply the conservation principle. When a solid is melted, its volume is the conserved quantity. The total volume of the original solid equals the total volume of the newly formed solid(s).

Step 3: Evaluate other properties. Surface area, edge length, and shape-specific properties (like curved surface area) change when the geometry changes. Only volume remains strictly conserved.

Answer: B

Question 8 · Spatial Aptitude MCQ

When a solid metal object is melted and recast into a completely different shape without any material loss or addition, which geometric property is strictly conserved?

A.
Total surface area
B.
Total volume
C.
Total edge length
D.
Curved surface area

Correct Answer:

Step-by-Step Solution

Key idea: This is a <conservation> question testing the fundamental principle of material transformation.

Step 1: Identify the transformation type. The keywords "melted and recast" indicate a change in state and shape, but the amount of matter remains exactly the same.

Step 2: Apply the conservation principle. The total volume of the original solid equals the total volume of the newly formed solid(s).

Step 3: Evaluate other properties. Surface area, edge length, and shape-specific properties change when the geometry changes. Only volume remains strictly conserved.

Answer: B

Question 9 · Spatial Aptitude MCQ

When a single straight cut is made completely through a solid, how does the total surface area of all the resulting pieces compare to the original solid's surface area?

A.
It remains exactly the same
B.
It increases by exactly the cross-sectional area of the cut
C.
It increases by exactly twice the cross-sectional area of the cut
D.
It decreases by the cross-sectional area of the cut

Correct Answer:

Step-by-Step Solution

Key idea: This is an <observation> question testing the surface area transformation rule after cutting.

Step 1: Visualize a single straight cut through a solid. The cut creates a new exposed surface on both of the resulting pieces.

Step 2: Identify the area of the new surfaces. Each new exposed surface has an area equal to the cross-sectional area of the cut.

Step 3: Calculate the total change. Since there are two new pieces, there are two new exposed surfaces. Therefore, the total surface area increases by exactly twice the cross-sectional area of the cut.

Answer: C

Question 10 · Spatial Aptitude MCQ

What is the minimum number of straight cuts required to divide a cube into 4 equal-sized pieces, if each cut must go completely through the cube and be parallel to one of its faces?

A.
2 cuts
B.
3 cuts
C.
4 cuts
D.
1 cut

Correct Answer:

Step-by-Step Solution

Key idea: This is a minimization question about cutting a cube into equal pieces, recognisable because it asks for the minimum number of cuts.

Step 1: Understand the constraints.

Cuts must be straight planes
Each cut goes completely through
Cuts are parallel to faces
We need 4 equal pieces

Step 2: Understand how cuts create pieces.

When cutting with planes parallel to faces:

n cuts in one direction create (n+1) pieces along that dimension
Total pieces = (pieces along x) × (pieces along y) × (pieces along z)

Step 3: Find how to get 4 pieces.

We need: (a) × (b) × (c) = 4

Possible factorizations:

4 × 1 × 1 (needs 3 cuts in one direction)
2 × 2 × 1 (needs 1 cut in x, 1 cut in y)

Step 4: Choose the minimum.

Option 1: 3 cuts (all in one direction)
Option 2: 1 + 1 = 2 cuts (orthogonal directions)

Minimum = 2 cuts

Step 5: Verify.

Cut 1: Slice horizontally → 2 pieces
Cut 2: Slice vertically (perpendicular to first) → 4 pieces

Answer: A (2 cuts)

three dimensional solids nets and orthographic views notes

Chapter Roadmap: Transformation of Shapes

Chapter Roadmap: Transformation of Shapes

What is a Cube Net

What is a Cube Net

Key Properties

Visual Intuition

Why This Matters

The Eleven Distinct Cube Nets

The Eleven Distinct Cube Nets

Common Categories

Recognition Strategy

Try a question

Login to view the complete notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Three-Dimensional Solids, Nets and Orthographic Views Notes for GATE CS

Chapter Roadmap: Transformation of Shapes

Chapter Roadmap: Transformation of Shapes

What is a Cube Net

What is a Cube Net

Key Properties

Visual Intuition

Why This Matters

The Eleven Distinct Cube Nets

The Eleven Distinct Cube Nets

Common Categories

Recognition Strategy

Method: Validating a Cube Net

Method: Validating a Cube Net

Step 1: Count the Squares

Step 2: Check Connectivity

Step 3: Quick Elimination Rules

Step 4: Mental Folding Test

Three-Dimensional Solids, Nets and Orthographic Views: Solved Questions with Step-by-Step Explanations (10 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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