Counting Figures

medium importance~0-1 Q in Tier 15 formulas⚡ 6 shortcuts3 subtopics

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Counting squares and rectangles

⏱ 4 min read🧩 5 question types🎯 16 practice Q

The idea in one minute

In a grid, count squares and rectangles by choosing lines, not by eye. Any 2 vertical grid lines and any 2 horizontal ones frame exactly one rectangle. For squares, go size by size: 1 by 1, then 2 by 2, and so on. Every square is also a rectangle, so rectangle counts include squares.

Every square is a rectangle

A rectangle has four sides and four right angles. A square is a rectangle with all four sides equal. So when a question asks for rectangles, the squares are inside the count. Only the phrase "rectangles that are not squares" leaves them out.

Count squares size by size

In a grid of m columns and n rows, count each size separately.

1 × 1 squares: $m\times n$
2 × 2 squares: $(m-1)(n-1)$
3 × 3 squares: $(m-2)(n-2)$ , and so on.

A 4 × 3 grid: $12+6+2=20$ squares. For a square n × n grid the total is $1^2+2^2+\dots+n^2$ . A chessboard: $1+4+9+\dots+64=204$ squares.

Tip: Stop when the square's side reaches the smaller of m and n. Bigger squares simply do not fit.

Count rectangles by picking lines

Never hunt shapes. A rectangle is fixed the moment you pick its two vertical sides and its two horizontal sides.

A grid with m columns has $m+1$ vertical lines. With n rows it has $n+1$ horizontal lines. Pick any 2 of each:

R=\binom{m+1}{2}\times\binom{n+1}{2}=\frac{m(m+1)}{2}\times\frac{n(n+1)}{2}

A 3 × 2 grid: $6\times3=18$ rectangles. A single row of m cells has $\dfrac{m(m+1)}{2}$ rectangles.

Rectangles that are not squares

Some questions ask for rectangles that are not squares. Count all rectangles, then subtract the squares: $R-S$ .

A 3 × 3 grid: $36-14=22$ rectangles that are not squares.

Watch: Options love to offer R and S alone. Read whether the question wants all rectangles or only the non-square ones.

Check with a small grid

When you forget a formula, test it on a tiny grid you can count by eye.

A 2 × 2 grid has 4 small squares and 1 big square, so 5 squares. It has 9 rectangles: 4 single cells, 4 two-cell blocks and 1 whole grid. The formula gives $3\times3=9$ , so it works.

Tilted squares

Join the middles of a square's four sides and you get a tilted square, a diamond. It counts as a square too.

A square with both midlines and the diamond: 4 small squares + 1 big square + 1 diamond = 6 squares. The midlines cut the diamond into 4 triangles, so no extra squares hide inside it.

Watch: A tilted square needs all four sides fully drawn. Grid lines crossing a diamond only make triangles.

Irregular grids

Plus signs, L shapes and grids with missing lines break the formulas. Check every candidate by hand.

Count the single cells.
For each bigger square or rectangle, check that all four sides are drawn.
Check the inside is completely filled.

Five equal squares in a plus sign: 5 single cells + 4 two-cell blocks + 1 full row + 1 full column = 11 rectangles.

Example: A missing middle line kills every rectangle that needs it as a side. Always trace the four sides.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Squares in a grid

How to spot it:

A chessboard-like grid; the question asks for squares of all sizes.

S=\sum_{k=1}^{\min(m,n)}(m-k+1)(n-k+1)

Method

Note m columns and n rows.
Add (m − k + 1)(n − k + 1) for k = 1, 2, ... up to the smaller side.
For an n × n grid use 1 squared + 2 squared + ... + n squared.

Why it works:

A k by k square fits in (m − k + 1) places across and (n − k + 1) places down.

Try this

How many squares are there in a 4 × 4 grid?

Show solution

$1^2+2^2+3^2+4^2$ .
$1+4+9+16$ .
$=30$ .

Answer

Type 2common4 practice Q

Rectangles in a grid

How to spot it:

A grid of cells; the question asks for rectangles (squares included).

R=\binom{m+1}{2}\binom{n+1}{2}

Method

Count the vertical lines (m + 1) and horizontal lines (n + 1).
Choose 2 of each set.
Multiply the two choose-2 values.

Why it works:

Choosing 2 vertical and 2 horizontal lines fixes exactly one rectangle.

Try this

How many rectangles are there in a grid of 4 columns and 3 rows?

Show solution

Vertical lines: 5, so $\binom{5}{2}=10$ .
Horizontal lines: 4, so $\binom{4}{2}=6$ .
$10\times6=60$ .

Answer

Type 3common3 practice Q

Squares with a tilted (diamond) square inside

How to spot it:

A square or grid with lines joining the middles of the sides, forming a diamond.

Method

Count the upright squares of the grid.
Add each tilted square whose four sides are fully drawn.
Do not count the triangles made where grid lines cut the diamond.

Why it works:

A tilted square is a square too, but grid lines through it only create triangles.

Try this

A square is divided into 4 equal squares by its two midlines, and the midpoints of its four sides are joined to form a diamond. How many squares are there?

Show solution

Upright: 4 small + 1 big = 5.
Tilted diamond: 1.
$5+1=6$ .

Answer

Type 4occasional3 practice Q

Irregular grids (plus, L-shape, missing lines)

How to spot it:

Cells joined in a cross or L shape, or a grid with a line missing.

Method

Count the single cells.
For each bigger shape, check all four sides are drawn.
For rectangles, fix a top-left cell and extend right and down while the region stays filled.

Why it works:

The grid formulas assume every line runs the full width; here they do not.

Try this

Five equal squares form a plus sign (one centre square, one on each side). How many rectangles are there?

Show solution

Single cells: 5.
Two-cell blocks (centre with an arm): 4.
Full row and full column: 2.
$5+4+2=11$ .

Answer

Type 5common

Rectangles that are not squares

How to spot it:

The question says 'rectangles which are not squares'.

\text{non-square} = R - S

Method

Count all rectangles with the line-pair method.
Count the squares size by size.
Subtract: non-square rectangles = R − S.

Why it works:

The full count includes squares, so removing the squares leaves exactly the rest.

Try this

How many rectangles in a 3 × 3 grid are not squares?

Show solution

All rectangles: $\binom{4}{2}^2=36$ .
Squares: $1+4+9=14$ .
$36-14=22$ .

Answer

Formula sheet

Rectangles in an m × n grid

R = \binom{m+1}{2}\binom{n+1}{2}

Pick 2 of the m + 1 vertical lines and 2 of the n + 1 horizontal lines.

Squares in an m × n grid

S = \sum_{k=1}^{\min(m,n)} (m-k+1)(n-k+1)

An n × n grid gives 1 squared + 2 squared + ... + n squared.

Shortcuts that save time

⚡ Line-pair trick

A rectangle is decided by its two vertical sides and two horizontal sides. Count line pairs, never shapes.

Example

How many rectangles are there in a 3 × 2 grid?

Show solution

Vertical lines: 4, so $\binom{4}{2}=6$ .
Horizontal lines: 3, so $\binom{3}{2}=3$ .
$6\times3=18$ .

Answer

⚡ Memorise the small grids

2 × 2: 5 squares, 9 rectangles. 3 × 3: 14 squares, 36 rectangles. 4 × 4: 30 squares, 100 rectangles. They appear inside bigger questions.

Example

How many squares are there in a 4 × 4 grid?

Show solution

$1^2+2^2+3^2+4^2$ .
$1+4+9+16$ .
$=30$ .

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Leaving squares out when the question asks for rectangles.

Every square is a rectangle. Subtract them only when the question says non-square.

Mistake 02

Using the number of cells instead of lines in the choose-2 count.

A grid with m columns has m + 1 lines. Choose 2 of the lines.

Mistake 03

Missing the bigger squares (2 × 2, 3 × 3) in a grid.

Count size by size up to the smaller side of the grid.

Mistake 04

Counting a tilted square's triangle pieces as more squares.

Grid lines through a diamond cut it into triangles, not squares.

Mistake 05

Applying the grid formula to a plus or L shape.

Irregular shapes need a side-by-side check: all four sides drawn, inside filled.

Quick revision

Read this the night before the exam.

Squares in an $m\times n$ grid: add $(m-k+1)(n-k+1)$ for each size k.
Square grid: $1^2+2^2+\dots+n^2$ ; chessboard = 204.
Rectangles: $\binom{m+1}{2}\times\binom{n+1}{2}$ .
Non-square rectangles = R − S.
Every square is a rectangle.
Tilted squares count once; irregular grids need a four-sides check.

Practice: 16 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 7 min · wrong answers go to your mistake notebook automatically.

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