How many rectangles or triangles.

Question

enter image description here

I have come across numerous questions where I am given the picture such as the above one been asked "how many rectangles are there?". I have even come across some slightly different images that instead of rectangles you are supposed to find the number of triangles. Well, I was thinking whether there is any formula or strategy that is used to solve these problems without having to manually count every shape.

Help would be appreciated.

Thank you :)

see: http://math.stackexchange.com/questions/429842/how-many-triangles-in-picture?rq=1 — Emilio Novati, Apr 18 '15 at 11:53
See also: Analysis of how-many-squares and rectangles are are there on a chess board? — Martin Sleziak, Aug 28 '18 at 08:19

score 10 · Accepted Answer · answered Apr 18 '15 at 11:59

10

To have a rectangle, you need 2 horizontal lines and 2 vertical lines. So for your given picture, there are $5\choose 2$ choices for two vertical lines. Also $4\choose 2$ choices for horizontal lines. So there are ${5 \choose 2}\times{4\choose2}$ rectangles in total.

The strategy is to find a way to categorize the things you want to count. Various problems will require various tricks, but you can gain experience by trying to solve them by your own.

answered Apr 18 '15 at 11:59

anumosh

945

What does the 5 over the 2 in brackets and 4 over the 2 in brackets mean. Is it just another way to show that it is a combination? – anonymous Apr 18 '15 at 12:06
@anonymous: It is called the binomial coefficient, which you can find at Wikipedia. – user21820 Apr 18 '15 at 12:08
So basically, what you're saying is that we can choose 2 of 5 possible x-coordinates 10 ways, and we can choose 2 from the 4 y-coordinates in 6 ways. Thus, this gives a total of 10×6=60 rectangles. Is that right? – anonymous Apr 18 '15 at 12:55
Right. That's what I meant. – anumosh Apr 18 '15 at 18:34
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In this case we are considering squares as rectangles aren't we? – kgkmeekg Apr 15 '16 at 05:08
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Oh so the mental picture is you cut exactly one rectangle out with 2 horizontal cuts and 2 vertical ones. – Raphael J.F. Berger Aug 28 '18 at 05:59

score 0 · Answer 2 · edited Oct 06 '20 at 17:27

0

I have found also this answer: $$\left(\sum_{i=1}^{\text{Length}+1} i\right)\cdot\left(\sum_{i=1}^{\text{Width}+1} i\right)$$

edited Oct 06 '20 at 17:27

Mohsen Shahriari

7,071

answered Oct 06 '20 at 15:47

Alex

103
1
1
8

How many rectangles or triangles.

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