Counting: Geometrical Patterns

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Counting: Shortcuts with Examples

Questions from this topic involve counting of geometrical figures such as squares, rectangles, triangles, parallelogram, quadrilaterals etc. in a given figure. We must count figures in a systematic method to save time. If we count the figures randomly, then we may make a mistake. So avoid counting figures randomly.

Let us consider some of the standard patterns and methods for counting the geometrical figures.

Consider a 4×4 square. Total number of squares formed can be counted as discussed below.

Number of 1 × 1 squares: 16 = 42
Number of 2 × 2 squares: 9 = 32
Number of 3 × 3 squares: 4 = 22
Number of 4 × 4 squares: 1 = 12
Total number of squares = 12 + 22 + 32 + 42 = (1 + 4 + 9 + 16) = 30

The formula can be generalized as under:
For a n×n square. Total number of squares formed are = 12 + 22 + 32 + .... + n2 = n(n + 1)(2n + 1)/6.

Example: For a 6×6 square
n = 6
Total number of squares formed = 6(6 + 1)(2×6 + 1)/6 = 91

Example: Find the total number of squares in the adjacent figure.


Given figure is a rectangle of size 3 × 4.
The smaller side is subdivided into 3 equal parts and the bigger side is subdivided into 4 equal parts, 12 smaller squares of equal size are obtained.
Now, the total number of squares in the figure can be counted as explained below:
Number of 1 × 1 squares = 3 × 4 = 12
Number of 2 × 2 squares = 2 × 3 = 6
Number of 3 × 3 squares = 1 × 2 = 2
Total number of squares formed = (3 × 4) + (2 × 3) + (1 × 2) = 12 + 6 + 2 = 20.

​Similarly, if a rectangle is of size 4 × 5 the total number of squares formed = (4 × 5) + (3 × 4) + (2 × 3) + (1 × 2) = 20 + 12 + 6 + 2 = 40

​Similarly, if a rectangle is of size 5 × 6 the total number of squares formed = (5 × 6) + (4 × 5) + (3 × 4) + (2 × 3) + (1 × 2) = 30 + 20 + 12 + 6 + 2 = 70

For a rectangle of dimension m × n the formula can be generalised as under
The total number of squares formed is equal to (m)×(n) + (m – 1)×(n – 1) + ... until one of the terms becomes zero.
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