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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What Do Shape Fitting Questions Look Like?
Students are given container dimensions and smaller tile measurements, and must calculate how many whole tiles fit without overlapping or cutting. Questions test row and column packing, handling leftover strips, and partitioning compound shapes. Here is a worked example.
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: Here, both regions fit whole tiles exactly. The area check agrees: (2 × 5) + (3 × 3) = 19 cm², covered by nineteen 1 cm² tiles.
Rules to Help You Solve Shape Fitting Questions
As shown in the example above, dividing dimensions into whole rows and columns ensures you pack tiles accurately. Use these core rules to count whole tiles, handle remainder strips, and partition compound shapes.
📐 Rule 1: Multiply Rows by Columns
For a rectangular region filled with tiles in one fixed orientation, divide each container dimension by the corresponding tile dimension. Round each result down to a whole number, then multiply the number across by the number down.
Example: 10 cm ÷ 2 cm = 5 shapes across, 6 cm ÷ 2 cm = 3 shapes down → 5 × 3 = 15 shapes.
✂️ Rule 2: Count Whole Tiles, Then Check Leftovers
A strip narrower than a tile cannot hold another tile on its own. For a simple rectangle, count only whole rows and columns. In a compound shape, check whether a leftover strip joins another usable region before discarding it.
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 → Count whole rows and columns to get 5 × 4 = 20 squares.
🧩 Rule 3: Partitioning Compound Shapes
Split compound shapes into non-overlapping rectangles, count the tiles in each, and add. A drawn partition is not a wall: tiles may cross it. If a split leaves narrow strips, move the partition or combine adjoining space to avoid losing places where whole tiles fit.
Example: Left column (10 squares) + Right foot (9 squares) = 19 total squares.
⚠️ Rule 4: Use Area as a Check
Container area divided by tile area gives an upper bound: round down to find a count you cannot exceed. It does not prove that many tiles fit. Area division gives the exact count when you have established that whole tiles cover the region without gaps or overlaps.
Example: For 2 cm squares in a 9 cm × 7 cm rectangle, 63 ÷ 4 = 15.75 gives an upper bound of 15 tiles. Checking whole rows and columns gives the actual fit: 4 × 3 = 12 squares.
Two More Worked Examples
Practise accounting for leftover strips when tiles do not divide evenly, then learn how to fit tiles across an imaginary partition.
Fitting Squares with Leftover Strips
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: Rules 1 and 2 — counting whole rows and columns in a rectangle.
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).
Fitting Tiles Across a Partition
Question: How many 2 cm × 2 cm square tiles fit into an L-shaped container with an overall width of 14 cm and height of 10 cm? Its left column is 9 cm wide, and its right foot is 5 cm high. Keep tile edges parallel to the container sides.
Options: A: 23 | B: 31 | C: 26 | D: 34 | E: 25
Worked Method
Step 1: Four columns of tiles fit in the left column: 9 ÷ 2 gives 4 whole tiles across. Five rows fit vertically: 10 ÷ 2 = 5. This places 4 × 5 = 20 tiles in an 8 cm × 10 cm block.
Step 2: This leaves a 1 cm strip inside the left column. At the bottom, that strip joins the right foot, whose width is 14 − 9 = 5 cm. The adjoining space is therefore 1 + 5 = 6 cm wide and 5 cm high.
Step 3: Fit 6 ÷ 2 = 3 tiles across this space and 2 whole rows down (5 ÷ 2 leaves 1 cm). That adds 3 × 2 = 6 tiles.
Answer: 20 + 6 = 26 tiles — option C.
Tip: Splitting at the 9 cm column edge and counting the 5 cm × 5 cm foot separately gives only 20 + 4 = 24. The partition is imaginary: letting tiles cross it uses the adjoining strip and fits two more tiles.
