Maths · Suggested Grade 5–6
Counting Rectangles in a Grid: Class 5–6 Visual Guide
Six little boxes can hide eighteen rectangles. The trick is to count bigger outlines too, without counting the same outline twice.
Original explanations and practice by BananaLearn · Published 1 November 2026
On a small screen, swipe across the illustration to read every label.
First, agree what counts
A rectangle has four right angles. In this guide, squares count as rectangles. Count only complete outlines whose sides lie along the horizontal and vertical grid lines. Do not include tilted shapes or invent missing lines. A 2-row, 3-column grid means six cells, not two horizontal and three vertical lines: it has three horizontal and four vertical boundary lines.
You need the properties of rectangles, multiplication and an organised list. Area asks how many unit squares cover a region. This puzzle asks how many different rectangular regions can be outlined. The six cells tell you the grid’s area in square units; they do not tell you the number of all rectangles.
Worked grid: one row and three columns
Draw three adjacent squares. There are three rectangles one cell wide: the left, middle and right cells. There are two rectangles two cells wide: left plus middle, and middle plus right. Finally there is one rectangle three cells wide: the whole row. The total is 3 + 2 + 1 = 6.
A repeatable rule is to fix a width before moving the outline. For a row of four cells, widths 1, 2, 3 and 4 give 4, 3, 2 and 1 positions, for 10 rectangles. Checking one width at a time makes a missing case visible.
Worked grid: two rows and three columns
Sort by height, then width. Height 1 and width 1 gives 2 × 3 = 6 positions. Height 1 and width 2 gives 2 × 2 = 4. Height 1 and width 3 gives 2 × 1 = 2. For height 2, widths 1, 2 and 3 give 3, 2 and 1 positions. Add 6 + 4 + 2 + 3 + 2 + 1 = 18.
The six cells are the smallest rectangles. The other twelve span more than one cell. A 1-by-2 shape and a 2-by-1 shape occupy different outlines in this grid, so both orientations belong in the list. The illustrated green outline identifies just one of the eighteen outlines.
A faster method: choose the four boundaries
Every grid-aligned rectangle has a left and right boundary chosen from the vertical lines and a top and bottom boundary chosen from the horizontal lines. In the 2-by-3 grid, there are four vertical lines. The distinct pairs are AB, AC, AD, BC, BD and CD: six choices. Three horizontal lines provide three pairs. Thus 6 × 3 = 18 rectangles.
Choose unordered pairs: AB and BA describe the same two boundaries. Two distinct lines are required; AA has zero width and is not a rectangle. Pairing each vertical choice with each horizontal choice counts every permitted rectangle exactly once. This explains the multiplication rather than merely announcing a formula.
Extension: a rule for any complete rectangular grid
For r rows and c columns, the number is [r(r + 1)/2] × [c(c + 1)/2]. The horizontal-line pairs are r + (r − 1) + ... + 1, and the vertical-line pairs are c + (c − 1) + ... + 1. A 3-by-4 grid therefore gives 6 × 10 = 60 rectangles. Use the listing method first if the algebra is new.
This rule assumes every grid segment is present. A missing internal segment may invalidate some outlines. It also excludes tilted rectangles. For an irregular diagram, return to checking actual closed boundaries; do not apply the complete-grid formula automatically.
What if the question excludes squares?
On a regular square-cell 2-by-3 grid, the squares are six 1-by-1 squares and two 2-by-2 squares: eight in total. Rectangles that are not squares therefore number 18 − 8 = 10. Write this interpretation explicitly if a puzzle uses the everyday phrase rectangles but not squares.
On a 3-by-3 square-cell grid, there are 9 + 4 + 1 = 14 squares. All rectangles number 6 × 6 = 36, leaving 22 non-square rectangles. These square counts require equal cell width and height; a grid drawn with stretched cells changes which outlines are geometrical squares.
Check a child’s method without pointing to the answer
Ask the learner to trace one outline, name its height and width in cells, and show where it belongs in the record. Then ask what changes when the outline moves one cell right. The aim is to connect a visible position to an entry, not to memorise eighteen.
A useful next challenge is to compare a 1-by-5 grid with a 2-by-3 grid. The first gives 15 rectangles and the second 18. More cells can create more rectangles, but cell count alone does not determine the answer: arrangement matters. If a learner double-counts, label boundary pairs rather than asking them to start the entire count again.
A complete 2-by-3 counting record
Copy the record into a notebook or use it to organise your own answer. Blank spaces are for your work.
| Height (cells) | Width (cells) | Positions | Your traced check |
|---|---|---|---|
| 1 | 1 | 6 | ___ |
| 1 | 2 | 4 | ___ |
| 1 | 3 | 2 | ___ |
| 2 | 1 | 3 | ___ |
| 2 | 2 | 2 | ___ |
| 2 | 3 | 1 | ___ |
| Total | All sizes | 18 | ___ |
12 questions with explained answers
Try the questions before opening the answers. Explain the reason, then try a changed example to check your understanding.
A complete 1-by-2 grid: how many rectangles, including squares?
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3
Two individual cells and the whole row.A 1-by-4 grid: how many?
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10
Widths 1 to 4 give 4 + 3 + 2 + 1.A 2-by-2 grid: how many?
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9
By dimensions: four 1-by-1, two 1-by-2, two 2-by-1 and one 2-by-2.How many width-2, height-1 outlines fit a 2-by-3 grid?
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4
Two horizontal positions in each of two rows.How many rectangles span the full height of a 2-by-3 grid?
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6
Choose any two of its four vertical lines: 3 + 2 + 1 pairs.A 3-by-3 grid: how many rectangles including squares?
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36
There are six boundary pairs in each direction; 6 × 6 = 36.A 3-by-4 grid: how many?
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60
Six horizontal-line pairs and ten vertical-line pairs.A 1-by-5 grid: how many?
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15
5 + 4 + 3 + 2 + 1 = 15.Exclude squares in a regular square-cell 2-by-3 grid. How many remain?
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10
Subtract eight squares from eighteen total rectangles.Exclude squares in a regular 3-by-3 square-cell grid. How many remain?
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22
36 − (9 + 4 + 1) = 22.Does choosing boundary pair AB and then BA create two rectangles?
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No
Their left and right boundaries are identical; order does not create a new outline.Can the complete-grid formula be used unchanged after deleting a segment?
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Not safely
An outline using the missing segment may no longer be closed. Check actual boundaries.
Keep the ideas connected
- Check what makes a rectangle
- Connect rows and columns to multiplication
- Compare counting all cases with sampling
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