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Counting of Geometrical Patterns

Questions in this chapter involve counting of geometrical figures such as squares, triangles, rectangles, parallelogram, etc., in a given figure. In order to count the figures accurately, we should follow a systematic method. If we adapt a random method for counting the figures, the chances of committing mistakes are high. Moreover, it may also result in wasting our valuable time.
 
We discuss the method below for counting the geometrical figures in some of the standard patterns.
 
Consider a square, which is divided into four parts horizontally as well as vertically. Total number of squares formed can be counted as under.
 
Description: 61764.png
 
Number of 4 × 4 squares: 1 = 12.
 
Number of 3 × 3 squares: 4 = 22.
 
Number of 2 × 2 squares: 9 = 32.
 
Number of 1 × 1 squares: 16 = 42.
 
Thus, total number of squares = 12 + 22 + 32 + 42.
 
= (1 + 4 + 9 + 16) = 30.
 
Suppose the square is divided into five parts on each side, then the total number of squares thus formed is equal to 12 + 22 + 32 + 42 + 52 = 55.
 
The formula can be generalised as under:
 
If a square is subdivided into ‘n’ parts equally on each side, then the total number of squares formed is:
12 + 22 + 32 + … + n2 Description: 61788.png.
 
If n = 5, total number of squares Description: 61804.png
 
Example-1
Find the total number of squares in the adjacent figure.
 
Description: 62633.png
 
Solution
The 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, thereby 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 = 12 = 3 × 4.
 
Number of 2 × 2 squares = 6 = 2 × 3.
 
Number of 3 × 3 squares = 2 = 1 × 2.
 
So, total number of squares = (3 × 4) + (2 × 3) + (1 × 2)
 
= 12 + 6 + 2 = 20.
 


​Similarly, if a rectangle of size 4 × 5 is subdivided into 20 squares of equal dimension, then the total number of squares is (4 × 5) + (3 × 4) + (2 × 3) + (1 × 2) = 20 + 12 + 6 + 2 = 40.
 
By extending this method to the rectangles of higher order, the formula can be generalised as under.
 
If a rectangle of dimension m × n is divided into ‘m.n’ number of smaller squares, then the total number of squares formed is equal to (m) (n) + (m – 1) (n – 1) + … until one of the terms becomes zero.
 
Example-2
Find the number of rectangles (including squares) in the adjacent figure.
 
Description: 61829.png
 
Solution
In the first horizontal strip ADHE, there are three smaller squares of size 1 × 1, two rectangles of size 2 × 1 (viz. ACGE and BDHF) and one rectangle of size 3 × 1 (viz. ADHE). In total, including squares, there are 3 + 2 + 1 = 6 rectangles.
 
There are four such horizontal strips. So, total number of rectangles, if counted taking one strip at a time is 4 × 6 = 24.
 
Now, consider two horizontal strips at a time.
 
There are three such strips, viz. ADLI, EHPM and ILTQ and in each strip there are six rectangles.
 
So, total number of rectangles in these strips = 3 × 6 = 18.
 
Then, consider the three strips at a time.
 
As there are two such strips and six rectangles in each strip, total number of rectangles in these strips = 2 × 6 = 12.
 
Finally, consider all the four strips together. There are six rectangles in this strip.
 
Thus, total number of rectangles in the figure = (4 × 6) + (3 × 6) + (2 × 6) + (1 × 6) = (4 + 3 + 2 + 1) × 6 = 10 × 6 = 60.
 
Total number of rectangles is 60.
 
The number rectangles excluding squares can be found as under:
 
Number of squares = 4 × 3 + 3 × 2 + 2 × 1 = 20.
 
Number of rectangles (including squares) = 4 × 6 + 3 × 6 + 2 × 6 + 1 × 6 = 60.
 
Hence, total number of rectangles excluding squares = 60 – 20 = 40.
 
 
Formula:
If a rectangle is divided into ‘n’ parts horizontally and ‘m’ parts vertically, total number of rectangles (including the squares) formed = Description: 61849.png.
 
Example-3
Find the number of triangles in the adjacent figure.
 
Description: 61864.png
 
Solution
First, count the number of triangles vertex wise.
 
From vertex 1, there are 4 triangles.
 
From vertices 2 and 3, there are 3 triangles each.
 
From vertices 4, 5 and 6, there are 2 triangles each.
 
Finally, from vertices 7, 8, 9 and 10, there is 1 triangle each.
 
Thus, total number of upright triangles = (4 × 1) + (3 × 2) + (2 × 3) + (1 × 4)
= 4 + 6 + 6 + 4 = 20.
 
From vertices 12, 13 and 14, there are 1 + 2 + 1 = 4 triangles formed in the inverted shape.
 
From vertices 8 and 9, there is 1 inverted triangle each and from vertex 5, there is 1 inverted triangle.
 
So, total number of inverted triangles = 4 + 2 + 1 = 7.
 
Hence, total number of triangles = 20 + 7 = 27.
 
 
Example-4
Count the number of triangles in the adjacent figure.
 
Description: 61874.png
 
Solution
Upright triangle = (5 × 1) + (4 × 2) + (3 × 3) + (2 × 4) + (1 × 5).
 
= 5 + 8 + 9 + 8 + 5 = 35.
 
Inverted triangles = (1 + 2 + 2 + 1) + (1 + 2 + 1) + (1 + 1) + 1 = 6 + 4 + 2 + 1 = 13.
 
Hence, total 35 + 13 = 48 triangles.
 
 
Example-5
Count the number of triangles in Figures (1), (2) and (3).
 
Description: 61888.png
 
Solution
Number of triangles in Figure (1) = 8.
 
Number of triangles in Figure (2) = 18.
 
Number of triangles in Figure (3) = 28.
 
 
Intersection of diagonals in a square, rectangle, quadrilateral, parallelogram, rhombus and trapezium will give eight triangles.





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