GMA Spatial Reasoning 2 — Questions and Answers

Question 1: A cube has been unfolded into a cross-shaped net with 6 squares. When re-folded, which pair of faces will be directly opposite each other?

Top and bottom only
Front and back only
Left and right only
All three pairs: top/bottom, front/back, left/right (Correct answer)

Correct answer: All three pairs: top/bottom, front/back, left/right

A cube always has exactly three pairs of opposite faces: top/bottom, front/back, and left/right. All six faces pair up into these three opposite pairs.

Spatial reasoning with cube nets requires understanding that a cube's 6 faces always form 3 pairs of opposites. No matter how the net is arranged, when folded, each face has exactly one face directly opposite it. The three pairs are: top-bottom, front-back, and left-right. This is a fundamental cube spatial relationship.

Question 2: Looking at a 3D object from above, you see a circle. From the front, you see a rectangle. What is the most likely 3D shape?

Sphere
Cylinder (Correct answer)
Cone
Cube

Correct answer: Cylinder

A cylinder viewed from above appears as a circle (the circular top face), and from the front appears as a rectangle (the curved side surface viewed straight on).

Interpreting 2D views of 3D objects is a core spatial reasoning skill. A cylinder has a circular cross-section (seen from above) and rectangular front profile (height × diameter). A sphere would show circles from all directions. A cone from above would show a circle but from the front a triangle. A cube shows squares from all directions.

Question 3: A rectangular piece of paper is folded in half twice (first horizontally, then vertically) and a hole is punched through all layers. How many holes appear when the paper is fully unfolded?

1
2
3
4 (Correct answer)

Correct answer: 4

Each fold doubles the number of layers. Folding twice creates 4 layers, so one punch creates 4 holes when unfolded.

Paper folding questions require tracking how folds multiply layers and how punched holes mirror across fold lines. After the first fold (horizontal): 2 layers. After the second fold (vertical): 4 layers. One hole punched through 4 layers creates 4 holes total. The holes are symmetrically arranged when unfolded.

Question 4: A 3×3×3 cube is assembled from 27 unit cubes. The entire outer surface is painted red. How many unit cubes have exactly 2 red faces?

8
12 (Correct answer)
6
1

Correct answer: 12

Unit cubes with exactly 2 red faces are the edge cubes (not corners). A 3×3×3 cube has 12 edges, each contributing one non-corner unit cube — giving 12 cubes with exactly 2 painted faces.

In a 3×3×3 painted cube: corners (8 cubes) have 3 painted faces; edge-centre cubes (12 cubes) have 2 painted faces; face-centre cubes (6 cubes) have 1 painted face; and the core cube (1) has 0 painted faces. Total: 8+12+6+1=27. The 12 edge-centre cubes each have exactly 2 faces painted.

Question 5: You are facing north. You turn 90° clockwise, then 180° anticlockwise. Which direction are you now facing?

North
South
East
West (Correct answer)

Correct answer: West

Start: North. Turn 90° clockwise → East. Turn 180° anticlockwise from East → West.

Mental rotation of direction requires applying changes sequentially. Starting from North: a 90° clockwise turn moves to East (N→E→S→W clockwise). From East, a 180° anticlockwise turn goes East → North → West, landing on West. Always apply rotations one step at a time to avoid errors.

Question 6: A shape is reflected across a vertical line and then rotated 180°. Which combination of operations produces the same result as a single reflection across a horizontal line?

Reflection across vertical + rotation 180° always equals horizontal reflection (Correct answer)
Never equivalent
Only if the shape is symmetric
Only for squares

Correct answer: Reflection across vertical + rotation 180° always equals horizontal reflection

Reflecting across a vertical axis and then rotating 180° is equivalent to reflecting across a horizontal axis — this is a fundamental property of isometries in the plane.

In transformation geometry, combining a vertical reflection with a 180° rotation always yields a horizontal reflection, regardless of the shape. This is because each transformation is an isometry, and their composition follows group multiplication rules. Reflection × (vertical) followed by rotation (180°) = reflection (horizontal). This identity holds universally.

GMA Spatial Reasoning 2

A cube has been unfolded into a cross-shaped net with 6 squares.
When re-folded, which pair of faces will be directly opposite each other?