Three-Dimensional Solids, Nets and Orthographic Views Short Notes for GATE CS
Three-Dimensional Solids, Nets and Orthographic Views short notes for GATE CS: 5 study cards covering concepts, formulas, shortcuts and exam traps, plus solve
three dimensional solids nets and orthographic views short notes
ConstraintCuts for equal rectangular pieces must be orthogonal and parallel to the original faces.
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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 nx planes parallel to the YZ-plane, ny planes parallel to the XZ-plane, and nz 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 Short Notes for GATE CS
Three-Dimensional Solids, Nets and Orthographic Views short notes for GATE CS: 5 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.
ConstraintCuts for equal rectangular pieces must be orthogonal and parallel to the original faces.
Exam Readiness: The Smooth Surface Rule
Exam Readiness: The Smooth Surface Rule
Rule 1A smooth 3D object has a unique, well-defined tangent plane at every single point.
Rule 2It cannot contain any sharp edges, corners, vertices, or cusps.
Rule 3If a question explicitly states an object is "smooth", immediately eliminate any option that describes or depicts edges, corners, or abrupt changes in surface direction.
Rule 4Surface texture (polished, rough, bumpy) is irrelevant to the mathematical definition of smoothness.
Three-Dimensional Solids, Nets and Orthographic Views: Solved Questions with Step-by-Step Explanations (10 Problems)
Question 1 · Spatial AptitudeMCQ
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:
B
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 AptitudeNAT
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:
7
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 AptitudeMCQ
If a solid cuboid is cut by nx planes parallel to the YZ-plane, ny planes parallel to the XZ-plane, and nz planes parallel to the XY-plane, what is the total number of pieces formed?
A.
nx+ny+nz
B.
(nx+1)(ny+1)(nz+1)
C.
nxnynz
D.
(nx+1)+(ny+1)+(nz+1)
Correct Answer:
B
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. nx cuts parallel to a plane divide the solid into nx+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 = (nx+1)×(ny+1)×(nz+1).
Answer: B
Question 4 · Spatial AptitudeMCQ
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:
C
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 AptitudeMCQ
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:
A
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 AptitudeMCQ
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:
B
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 AptitudeMCQ
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:
B
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 AptitudeMCQ
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:
B
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 AptitudeMCQ
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:
C
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 AptitudeMCQ
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:
A
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