Spatial Reasoning

Spatial reasoning is the ability to think about shapes and space in your head. It is the core skill behind every NVR question type, but some questions test it directly. These include questions about hidden shapes, embedded figures, counting shapes, and fitting shapes together (like a jigsaw puzzle).

This lesson brings together all the spatial skills you have been developing and introduces some new question types.

---

Type 1: Hidden Shapes (Embedded Figures)

In these questions, you are shown a simple shape and asked to find it hidden inside a more complex figure.

How It Works

Target shape: A small right-angled triangle

Complex figure: A geometric pattern made of many overlapping lines and shapes.

Question: Which answer option shows the target shape correctly embedded in the complex figure?

Strategy for Hidden Shapes

1. Study the target shape carefully. Note its exact proportions, angles, and size.
2. Look at the complex figure. Try to find lines or edges that match the target shape's sides.
3. Check the orientation. The hidden shape might be rotated but it should NOT be reflected (unless the question says otherwise).
4. Verify all sides. Every side of the target shape must lie exactly along a line in the complex figure.

Worked Example

Target shape: A rectangle (wider than it is tall)

Complex figure:
     /\
    /  \
   /----\
  / |  | \
 /  |  |  \
/___+--+___\

The complex figure looks like a large triangle with a rectangle drawn inside it.

The hidden rectangle is the shape formed by the vertical and horizontal lines inside the triangle. Even though the triangle dominates the image, the rectangle is there if you look for it.

---

Type 2: Counting Shapes

These questions ask: "How many triangles (or squares, rectangles, etc.) can you find in this figure?"

Strategy for Counting Shapes

The trick is that shapes can overlap and combine. A figure that looks like it has 2 small triangles might actually have 3 — the two small ones plus one large triangle made up of both.

Systematic approach:

1. Count the smallest shapes first (the individual units).
2. Then count combinations of 2 small shapes that form a larger version.
3. Then count combinations of 3, and so on.
4. Add them all up.

Worked Example

Figure: A large triangle divided into 4 smaller triangles by connecting the midpoints of each side.

        /\
       /  \
      /    \
     /------\
    /\      /\
   /  \    /  \
  /    \  /    \
 /------\/------\

Count:

• Small triangles pointing up: 3 (top, bottom-left, bottom-right)
• Small triangle pointing down: 1 (the inverted one in the middle)
• Medium triangles (made of 4 small triangles): 3 (left half, right half, bottom half)

Wait — let us be more careful:

• Medium triangle (top-left + top-right small triangles + inverted middle) — this does not form a triangle shape.

Let me recount systematically:

• Individual small upward triangles: 3
• Individual small downward triangle: 1
• Triangle made of top small triangle + inverted middle: Not a triangle
• Triangle made of top + bottom-left + inverted middle: this creates the LEFT medium triangle = 1
• Right medium triangle: 1
• Bottom medium triangle (bottom-left + inverted + bottom-right): 1
• The whole large triangle: 1

Total: 3 + 1 + 3 + 1 = 8 triangles (but verify by listing them).

Top Tip: When counting, label each small section (A, B, C, D...) and then list all possible combinations that form the target shape. This prevents you from missing any or counting one twice.

---

Type 3: Fitting Shapes Together (Jigsaws)

These questions give you a shape with a piece missing and ask which piece completes it, or they ask which pieces fit together to make a given shape.

Strategy

1. Look at the outline of the missing space. What shape is the gap?
2. Check the size. The piece must fit exactly — not too big, not too small.
3. Check the orientation. The piece might need to be rotated to fit.
4. Check internal details. If the shape has a pattern, the piece must continue that pattern correctly.

Worked Example

A square is divided into 4 quarters. The top-right quarter is missing.

Full square:
+-----+-----+
|     |  ?  |
|  A  |     |
+-----+-----+
|     |     |
|  C  |  D  |
+-----+-----+

Quarter A has horizontal stripes.
Quarter C has vertical stripes.
Quarter D has diagonal stripes.

What pattern should the missing quarter (?) have?