Shape Fitting Practice Test Worksheets
Build confidence with printable shape fitting worksheets and focused spatial reasoning practice tests. Students learn to fit smaller squares and rectangles into larger rectangles, compound shapes, and hollow shapes while improving accuracy with visual packing problems.
For students who can follow basic measurements, but need targeted practice with rows, columns, leftover space, and compound-shape partitioning.
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The Rules Every Shape Fitting Question Uses
Shape fitting puzzles measure visual packing efficiency. Mastery comes down to 4 fundamental principles:
📐 Rule 1: Row × Column Dimensional Packing
Calculate how many whole small shapes fit horizontally (Container Length ÷ Shape Length) and vertically (Container Width ÷ Shape Width), then multiply them together.
Example: 10 cm ÷ 2 cm = 5 shapes across, 6 cm ÷ 2 cm = 3 shapes down → 5 × 3 = 15 shapes.
✂️ Rule 2: Integer Division & Ignore Remainder Space
The smaller shapes cannot be cut into pieces. Any leftover strip smaller than a full shape dimension must be ignored for that row or column.
Example: For an 11 cm × 9 cm box with 2 cm squares: 11 cm ÷ 2 cm = 5 (rem 1 cm) across, and 9 cm ÷ 2 cm = 4 (rem 1 cm) down → Discard remainders to get 5 × 4 = 20 squares.
🧩 Rule 3: Partitioning Compound Shapes
Break L-shapes, U-shapes, plus-shapes, and hollow rectangles into simple non-overlapping rectangles. Count the small shapes in each sub-region and sum them.
Example: Left column (10 squares) + Right foot (9 squares) = 19 total squares.
⚠️ Rule 4: Dimensional Packing vs Area Division Trap
Do not rely on total area division when dimensions leave remainders. Whole shape dimensions along each axis dictate physical fit.
Example: 9 cm × 7 cm = 63 cm². Dividing by 4 cm² gives 15.75, but true packing is 4 × 3 = 12 squares.
How to Solve Shape Fitting Questions
One compound shape partition, one dimension packing with remainder, and one classic area trap.
1. Fitting Squares into a Compound L-Shape
Question: How many 1 cm × 1 cm square tiles can fit into the L-shaped container shown below? (Bounding box is 5 cm × 5 cm, left column width is 2 cm, right foot height is 3 cm).
Options: A: 22 | B: 16 | C: 19 | D: 24 | E: 17
Worked Method
What it's testing: Rule 3 — partitioning compound L-shapes into simple rectangular components.
Step-by-step evaluation:
- Left column: Dimensions are 2 cm × 5 cm → (2 ÷ 1) × (5 ÷ 1) = 2 × 5 = 10 tiles.
- Right foot: Dimensions are 3 cm × 3 cm → (3 ÷ 1) × (3 ÷ 1) = 3 × 3 = 9 tiles.
- Total tiles: 10 + 9 = 19 tiles.
Answer: C (19)
Tip: Always draw a horizontal or vertical line to divide compound shapes into clean non-overlapping rectangular blocks before calculating tile counts.
2. Dimension Packing with Leftover Remainder Space
Question: How many 2 cm × 2 cm square tiles can fit into a rectangle measuring 11 cm by 9 cm?
Options: A: 19 | B: 18 | C: 20 | D: 23 | E: 9
Worked Method
What it's testing: Rule 1 & Rule 2 — integer axis division and discarding remainder space.
Step-by-step evaluation:
- Horizontal fit: 11 cm ÷ 2 cm = 5 tiles (with 1 cm leftover remainder).
- Vertical fit: 9 cm ÷ 2 cm = 4 tiles (with 1 cm leftover remainder).
- Total tiles: 5 × 4 = 20 tiles.
Answer: C (20)
Tip: Option E (9) comes from mistakenly adding 5 + 4 instead of multiplying rows by columns (5 × 4 = 20).
3. Avoiding the Area Division Trap
Question: How many 2 cm × 2 cm square tiles fit into a rectangle measuring 9 cm by 7 cm?
Options: A: 12 | B: 13 | C: 20 | D: 15 | E: 17
Worked Method
What it's testing: Rule 4 — avoiding area division errors.
Step-by-step evaluation:
- Horizontal fit: 9 cm ÷ 2 cm = 4 tiles (remainder 1 cm).
- Vertical fit: 7 cm ÷ 2 cm = 3 tiles (remainder 1 cm).
- True dimensional packing: 4 × 3 = 12 tiles.
- The Trap (Option D = 15): Total area = 9 × 7 = 63 cm². Tile area = 2 × 2 = 4 cm². 63 ÷ 4 = 15.75 → 15. This assumes leftover 1 cm strips can be melted together, which is physically impossible!
Answer: A (12)
Tip: Always calculate axis fits individually before multiplying. Never rely on area division!
Before You Buy
Want to check the level and layout first? Download the free sample PDF. It uses the same question style, printable format, and answer-key approach as the full pack.
Download Free Sample PDFGet the Full Practice Pack
The full Shape Fitting pack contains 90 questions across 3 printable test sets. Students practise spatial reasoning, rectangular packing, compound and hollow shapes, and remainder space calculation.
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