How To Count Quadrilaterals!

Hello again fellow quadrilateral enthusiasts!

Today, we're here to learn how to count quadrilaterals, so sit back, relax, and go grab a cup of coffee if you want to (or tea, if you prefer).

Just in case you don't know, quadrilateral is a polygon that has 4 sides.

Again let's start off with an example:

How many quadrilaterals are there in the 6×46\times4 grid below?

Of course we could go old school and count them one by one, but we're gonna go the smarter way.

To construct a quadrilateral, we need two horizontal lines, and two vertical lines.

In the 6×46\times4 grid above, there are 7 vertical lines and 5 horizontal lines, we choose 2 from each of them.

Hence, the total number of quadrilaterals in a 6×46\times4 grid is (72)×(52)=210{7\choose 2} \times {5\choose 2} =210


Now, generally, how do we count the number of quadrilaterals in an a×ba\times b grid (where aa is the width of the grid and bb is the height of the grid)?

Similarly, to construct a quadrilateral, we need two horizontal lines and two vertical lines.

In an a×ba\times b grid, there are a+1a+1 vertical lines and b+1b+1 horizontal lines, we choose two from each of them.

So, the number of quadrilaterals in an a×ba\times b grid is (a+12)×(b+12)=ab(a+1)(b+1)4{a+1\choose 2}\times{b+1\choose 2}=\frac{ab(a+1)(b+1)}{4}

The number of quadrilaterals in an a×ba\times b grid (where aa is the width of the grid, bb is the height of the grid) is ab(a+1)(b+1)4\frac{ab(a+1)(b+1)}{4}


This is one part of Quadrilatorics.

Note by Kenneth Tan
3 years, 4 months ago

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A complex problem has been beautifully simplified. Thanks a lot.

Niranjan Khanderia - 3 years, 3 months ago

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Pro.. .

Yap Zhen Hao - 3 years, 1 month ago

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I beg to disagree... What if the sides of the quadrilateral are not parallel to the grid?

You cannot count them using this method...

In this method, you can only count squares and rectangles

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