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

Solve 5+ Three-Dimensional Solids, Nets and Orthographic Views previous year questions for GATE CS with answers and detailed solutions. Free sample questions below.

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.

Three-Dimensional Solids, Nets and Orthographic Views: Solved Questions with Step-by-Step Explanations (5 Problems)

Question 1 · Spatial Aptitude · 2026_Set1 MCQ

In Panel I of the figure below, the front view and top view of a structure are shown. Which one of the 3D structures shown in Panel II possesses the views shown in Panel I?

A.
(i)
B.
(ii)
C.
(iii)
D.
(iv)

Correct Answer:

Step-by-Step Solution

Key idea: This is a <pattern_matching> question requiring 3D reconstruction from orthographic views. The task is to match the given front and top views to the correct 3D structure.

Step 1: Analyze the Front View.

The front view shows a stepped pattern with 3 levels:

Left column: 3 blocks high
Middle section: 2 blocks high (shifted right)
Right section: 2 blocks at bottom level

Step 2: Analyze the Top View.

The top view shows a 3×3 grid pattern:

Top row: 3 blocks
Middle row: 3 blocks
Bottom row: 2 blocks (left and middle only)

Step 3: Reconstruct the 3D shape mentally.

Combining both views:

The structure has depth (visible from top view showing 3 rows)
The left side is tallest (3 levels from front view)
There's an asymmetrical stepped pattern

Step 4: Match with Panel II options.

Option (i): Shows correct stepped pattern with proper depth

Option (ii): Missing the middle step feature

Option (iii): Incorrect depth arrangement

Option (iv): Wrong orientation of steps

Answer: C (Option iii matches the views correctly)

Question 2 · Spatial Aptitude · 2025_Set2 MCQ

The paper as shown in the figure is folded to make a cube where each square corresponds to a particular face of the cube. Which one of the following options correctly represents the cube?

Note: The figures shown are representative.

Correct Answer:

Step-by-Step Solution

Key idea: This is a <net_folding> question testing ability to visualize cube formation from a 2D net. We must determine which 3D cube configuration correctly represents the folded net.

Step 1: Analyze the given net.

The net shows 6 squares in a cross pattern:

Top square: Black circle
Middle row (left to right): White square, White triangle, Black triangle, White square
Bottom square: White circle

Step 2: Identify opposite faces.

In a cube net, faces separated by one square in a straight line are opposite:

Black circle (top) is opposite to White circle (bottom)
Left white square is opposite to Black triangle
White triangle is opposite to Right white square

Step 3: Identify adjacent faces.

The White triangle (center) is adjacent to:

Black circle (above)
White circle (below)
Left white square (left)
Black triangle (right)

Step 4: Check each option.

Option A: Shows circle on top, triangle on front - need to verify if this matches adjacency

Option B: Shows two triangles adjacent - check if valid

Option C: Shows triangle on front, correct orientation

Option D: Shows triangle and circle in wrong positions

Step 5: Verify Option C.

Looking at the visible faces in Option C:

Front face: White triangle (matches center of net)
Top face: Consistent with folding
Right face: Consistent with net arrangement

Answer: C

Question 3 · Spatial Aptitude · 2024_Set2 MCQ

A cube is to be cut into 8 pieces of equal size and shape. Here, each cut should be straight and it should not stop till it reaches the other end of the cube.

The minimum number of such cuts required is

A.
3
B.
4
C.
7
D.
8

Correct Answer:

Step-by-Step Solution

Key idea: This is a <optimization> question about minimizing cuts to subdivide a cube into equal pieces. The key is understanding how orthogonal cuts multiply the number of pieces.

Step 1: Understand the cutting constraint.

Each cut must be straight
Each cut goes completely through the cube
Cuts are parallel to faces (to get equal pieces)

Step 2: Understand how cuts create pieces.

When cutting a cube with planes parallel to faces:

n cuts parallel to one face create (n+1) pieces along that dimension
Total pieces = (cuts_x + 1) × (cuts_y + 1) × (cuts_z + 1)

Step 3: Find the factorization of 8.

We need 8 equal pieces.

8 = 2 × 2 × 2

This means we need:

2 pieces along x-axis → 1 cut
2 pieces along y-axis → 1 cut
2 pieces along z-axis → 1 cut

Step 4: Calculate minimum cuts.

Total cuts = 1 + 1 + 1 = 3 cuts

Step 5: Verify this is minimum.

Alternative: Could we use fewer cuts?

With 2 cuts maximum: Best case is 2 cuts in different directions
This gives at most 2 × 2 × 1 = 4 pieces (not enough)
With 3 cuts in 3 orthogonal directions: 2 × 2 × 2 = 8 pieces ✓

Therefore, 3 cuts is the minimum.

Step 6: Visualize the solution.

Cut 1: Slice horizontally through middle → 2 pieces
Cut 2: Slice vertically (front-to-back) through middle → 4 pieces
Cut 3: Slice vertically (left-to-right) through middle → 8 pieces

Answer: A (3 cuts)

Question 4 · Spatial Aptitude · 2024_Set1 MCQ

A rectangular paper sheet of dimensions

54 cm \times 4 cm

is taken. The two longer edges of the sheet are joined together to create a cylindrical tube. A cube whose surface area is equal to the area of the sheet is also taken.

Then, the ratio of the volume of the cylindrical tube to the volume of the cube is

A.
$1/ π$
B.
$2/ π$
C.
$3/ π$
D.
$4/ π$

Correct Answer:

Step-by-Step Solution

Key idea: This is a <conservation> question involving shape transformation with conserved area. The rectangular sheet transforms into a cylinder (area conserved as curved surface area) and we compare with a cube of equal surface area.

Step 1: Calculate the sheet area.

Sheet dimensions: 54 cm × 4 cm

Area = 54 × 4 = 216 cm²

Step 2: Form the cylinder by joining longer edges.

When longer edges (54 cm) join:

Circumference = 54 cm
Height of cylinder = 4 cm

Step 3: Find cylinder radius.

Circumference = 2πr = 54

r = 54/(2π) = 27/π cm

Step 4: Calculate cylinder volume.

V_cyl = πr²h = π × (27/π)² × 4

V_cyl = π × (729/π²) × 4

V_cyl = 2916/π cm³

Step 5: Find cube dimensions from surface area.

Cube surface area = Sheet area = 216 cm²

6a² = 216 (where a = side of cube)

a² = 36

a = 6 cm

Step 6: Calculate cube volume.

V_cube = a³ = 6³ = 216 cm³

Step 7: Find the ratio.

Ratio = V_cyl / V_cube

Ratio = (2916/π) / 216

Ratio = 2916/(216π)

Ratio = 13.5/π

Checking options: 13.5/π ≈ 4.297/π

Wait, let me recalculate:

2916/216 = 13.5

But options are 1/π, 2/π, 3/π, 4/π

Let me reconsider: perhaps the shorter edges join?

If 4 cm edges join:

Circumference = 4 cm
Height = 54 cm
r = 4/(2π) = 2/π cm
V_cyl = π × (2/π)² × 54 = π × 4/π² × 54 = 216/π cm³
Ratio = (216/π)/216 = 1/π

Answer: A

Question 5 · Spatial Aptitude · 2023 MCQ

Looking at the surface of a smooth 3-dimensional object from the outside, which one of the following options is TRUE?

A.
The surface of the object must be concave everywhere.
B.
The surface of the object must be convex everywhere.
C.
The surface of the object may be concave in some places and convex in other places.
D.
The object can have edges, but no corners.

Correct Answer:

Step-by-Step Solution

Key idea: This is a <conceptual> question testing understanding of surface curvature and the mathematical definition of smooth surfaces in 3D geometry.

Step 1: Understand what "smooth" means mathematically.

A smooth 3D surface has:

A unique tangent plane at every point
No sharp edges or corners
Continuous curvature (differentiable everywhere)

Step 2: Analyze each option.

Option A: "Must be concave everywhere"

FALSE: A sphere is smooth and convex everywhere
A smooth surface can be convex, concave, or mixed

Option B: "Must be convex everywhere"

FALSE: A smooth torus (donut shape) has both concave and convex regions
The inner part is concave, outer part is convex

Option C: "May be concave in some places and convex in other places"

TRUE: This is correct
Example: A torus, or a wavy surface, or an ellipsoid with varying curvature
Smoothness only requires continuous differentiability, not uniform curvature type

Option D: "Can have edges, but no corners"

FALSE: A smooth surface cannot have edges
Edges represent discontinuities in the tangent plane
By definition, smooth surfaces have no edges or corners

Step 3: Conclusion.

A smooth 3D object can have varying curvature - concave in some regions, convex in others - as long as the surface remains differentiable everywhere.

Answer: C

Try a question

Login to view the complete previous-year questions and solutions

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 PYQs 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

Three-Dimensional Solids, Nets and Orthographic Views: Solved Questions with Step-by-Step Explanations (5 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

More previous year questions (pyqs) in this unit