Permutations and Combinations
8. Application in Geometry
8.2. Squares, Rectangles and Parallelograms
In a rectangular or square grid, the number of squares (and rectangles) shows a pattern. Using this pattern, we can develop a formula to calculate number of squares and rectangles in the grid. Let us consider the grid of size 3×3.
Hence total number of squares = 9 + 4 + 1 = 13.
Take one more example:Consider the following grid of size 6×4.
Number of squares in the grid of all possible size:
Number
of squares of side 1 = 4 × 6
Number
of squares of side 2 = 3 × 5
Number
of squares of side 3 = 2 × 4
Number
of squares of side 4 = 1 × 3
Total number of squares = 24 +15 + 8 + 3 = 50
In a grid of dimension \(m \times n,\;m > n\) total number of squares of all possible dimensions
= \(mn + (m - 1)(n - 1) + (m - 2)(n - 2) + .....\)
When \(m = n\), then number of squares is:
\( = {m^2} + {(m - 1)^2} + {(m - 2)^2} + ....... + {2^2} + {1^2}\)
Number of rectangles in the same grid:
To form a rectangle, we need two vertical lines and two horizontal lines. In the given grid, there are \(m + 1\) horizontal lines and \(n + 1\) vertical lines and by selecting any two vertical and any two horizontal lines we get a rectangle
Number of rectangles = \(^{m + 1}{{\rm{C}}_2} \times {\,^{n + 1}}{{\rm{C}}_2}\)
Number of parallelograms is same as number of rectangles.
Example 01: Find the number of squares of all possible dimensions in the given grid.
Number of squares in the inner grid is also 14. But one square is common that is counted twice. Hence the required number of squares = 14 + 14 – 1 = 27.