Many of you wanted shortcuts to count the number of triangles. So I am writing this post which will cover every possible scenario for counting squares, rectangles and triangles. If the question doesn't fall in any of the below category, you will have to count manually :)

How many squares are there in the above figure?

Apply the formula [n(n+1)(2n+1)/6]

= 4*5*9/6

So total number of squares is

Here n = 4, m = 3

How many rectangles are there in the above figure?

In the above figure

n = 4, m=6

Put the values

The formula to count the number of rectangles within a square can be obainted by putting n = m in the above formula (3rd case)

Put n = 8 in the above formula

1. It is only applicable for matchstick arrangement (like the figure above)

2. n = number of unit triangles in a side. In this question n=4 (I have counted the triangles for you)

3. Those are not brackets enclosing the formula; they denote floor function. So after applying the formula, if you are not getting an integer, you should completely reject the decimal numbers. E.g. 17.12 will become 17 and 56.98 wil become 56

Put n=4 in the formula

Here n=3

Apply the formula and you will get 13.125. Reject the decimal places-

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**1)**__Counting squares within a square__How many squares are there in the above figure?

**Solution**: There are 4 rows in the above figure. Hence n=4Apply the formula [n(n+1)(2n+1)/6]

= 4*5*9/6

**Answer : 30**

**2)**__Counting squares within a rectangle__

**How many squares are there in the above figure?**

**Solution**: This rectangle is a 4 x 3 grid [4 rows and 3 columns]So total number of squares is

**m(m+1)(2m+1)/6 + (n-m)*m(m+1)/2.**where n is the larger dimensionHere n = 4, m = 3

**Answer : 20**

**3)**__Counting rectangles within a rectangle__

How many rectangles are there in the above figure?

**Solution**: For a 'n x m' grid ['n' is the number of rows and 'm' is the number of columns]**Number of rectangles = (m)(m+1)/2 (n)(n+1)/2 = m(m+1)(n)(n+1)/4**In the above figure

n = 4, m=6

Put the values

**Answer : 210**

**4)**__Counting rectangles in a square__The formula to count the number of rectangles within a square can be obainted by putting n = m in the above formula (3rd case)

**So, No. of rectangles = [n(n+1)/2]^2**

**Ques.**How many rectangles are there in a chess board?

**Solution**: We know that the chess board is a 8 x 8 gridPut n = 8 in the above formula

**Answer : 1296**

**Note : From the above formulae, you only have to mug two, i.e., 2 and 3. Formula 1 can be obtained by putting n=m in formula 2 and formula 4 can be obtained by putting n=m in formula 3.**

**5)**__Counting triangles within a triangle__

**How many triangles are there in the above figure?**

The formula is given by -The formula is given by -

**You should note 3 things about this formula-**1. It is only applicable for matchstick arrangement (like the figure above)

2. n = number of unit triangles in a side. In this question n=4 (I have counted the triangles for you)

3. Those are not brackets enclosing the formula; they denote floor function. So after applying the formula, if you are not getting an integer, you should completely reject the decimal numbers. E.g. 17.12 will become 17 and 56.98 wil become 56

*Caution :*You don't have to round off the number, so 77.9 will be equal to 77 not 78.Put n=4 in the formula

**Answer : 27**__Next question-__Here n=3

Apply the formula and you will get 13.125. Reject the decimal places-

**Answer : 13**

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couldn't undetsyand d logic behind these formulae

ReplyDeletecoz i lv to find out d logic behind evry formula, which makes it easy for me mug them up

anyways thanx for d info

waiting for ur next post

and please if it possible for u to speed up a li'l bit

that will b highly appreciated

Thanx n regards

couldn't undetsyand d logic behind these formulae

ReplyDeletecoz i lv to find out d logic behind evry formula, which makes it easy for me mug them up

anyways thanx for d info

waiting for ur next post

and please if it possible for u to speed up a li'l bit

that will b highly appreciated

Thanx n regards

if a=9 b=8 and c=-17 then find the value of (a^4+b^4+c^4)/(a^2*b^2+b^2*c^2+c^2*a^2)=???

ReplyDeleteA B C can do a piece of work in 6 hours working together. If A and C working together works four times of B's work and A working alone works 5 times of C's work then only A will finish the work in??

ReplyDelete9 days

Deleteur answer is right any short cut of this u knw then post the solution

Delete1/a + 1/b + 1/c = 1/6 ...(1)

Delete1/a + 1/c = 4/b ...(2)

1/a = 5/c or c=5a ...(3)

Put (2) in (1)

5/b = 1/6

b = 30 days ...(4)

Put (3) and (4) in (2)

1/a + 1/5a = 4/30

6/5a = 2/15

or a = 9 days

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ReplyDelete1. 40!^40!= ?

2. 2!^2!+4!^4!+6!^6!+8!^8!+10!^10!= ?

thanks a lot for the above formula

ReplyDeleteI think formula for model 3 is wrong

ReplyDeletehere is the video for counting of figures: https://youtu.be/4RM8Fh5Jx0M

ReplyDeleteWe can't remember every formula.. You should provide some better ways to solve those (not formulae)

ReplyDeleteHow would you count rectangles in the chess board if the centre 4x4 grid were cut out of the middle of it?

ReplyDeleteThank you for proving good information to count triangles easily.

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ReplyDeletePlz plz sir