Calculate the number of triangles in a square

Type 1 : Counting triangles with in Square, Rectangle, Quadrilateral 

Figure – 1 : Number of triangles in Fig – 1 =  8
Hint: Here having total  two diagonals and having four blocks. So formula for that 4 x 2 = 8 number of triangles.

Figure – 2 : Number of triangles in Fig – 2  =  16
Hint: Here having total  two diagonals and having eight blocks. So formula for that 8 x 2 = 16 number of triangles.

Figure – 3 : Number of triangles in Fig – 3  =  18
Hint: Here each square having 8 no. of triangles and combine squares having 2 no. of triangles. So total number of   triangles – 8 + 8 + 2 = 18.

Figure – 4 : Number of triangles in Fig – 3  =  28
Hint: Here each square having 8 no. of triangles and combine squares having 4 no. of triangles. So total number of   triangles – 8 + 8 + 8 + 4 = 28.

Type – 2 : Counting triangles with the Triangle having number of bisects with vertex

Figure – 5: Number of possible triangles in Fig – 5 =  1

Figure – 6 : Number of possible triangles in Fig – 6 =  3

Formula : Here number of parts ” n”  then possible triangles  is n (n+1) /2

Figure – 7 : Number of possible triangles in Fig – 7 =  10
Hint : No of parts ” n” = 4 so according to formula 4 x 5 /2 = 10

Figure – 8 : Number of possible triangles in Fig – 8 =  15
Hint : No of parts ” n” = 5 so according to formula 5 x 6 /2 = 15.

Type – 3 : Counting triangles with the Triangle having number of bisects with vertex and horizontal lines

Figure – 9: Triangle counting in Fig – 9 =  2

Figure – 10: Triangle counting in Fig – 10 =  6

Formula : Here number of vertical parts ” n” and horizontal parts “m” then possible triangles  is   nxm(n+1)/2

Figure – 11: Triangle counting in Fig – 11 =  30
Solution : Here number of vertical parts ” 4″ and horizontal parts “3” then possible triangles  is 4 x 3 x 5 /2 = 30

Figure – 12: Triangle counting in Fig – 12 =  45
Solution : Here number of vertical parts ” 5″ and horizontal parts “3” then possible triangles  is 5 x 3 x 6 /2 = 45

Type – 4 : Counting triangles with in embedded Triangle

Figure – 13: Triangle counting in Fig – 13 =  5

Formula : Here number embedded triangles in outer triangle ” n” and horizontal parts “m” then possible triangles  is 4n + 1

Figure – 14: Triangle counting in Fig – 14 =  9 ( Here n= 2 )

Figure – 15: Triangle counting in Fig – 15 =  13 ( Here n= 3 )

Type – 5 : Counting triangles with in the  particular pattern of Triangle

Formula to count number of triangles like above particular pattern type of Triangle ={n (n+2) (2n+1)}/8   where  “n”  = number of unit triangles in a side

Note : Consider only integer part from answer obtained in above formula ( For example the answer may come 13.12 then consider only “13”. Also remember You don’t have to round off the number for example answer may come 36.8 then consider only “36”.

Figure – 16: No of triangles in Fig – 16 =  13 ( Here n= 3 )
Solution: According to above formula 3 x 5 x 7 /8 = 13.12 so consider integer only i.e 13

Figure – 17: No of triangles in Fig – 17 =  27 ( Here n= 4 )
Solution: According to above formula 4 x 6 x  9/8 = 27

Figure – 18: No of triangles in Fig – 18 =  170 ( Here n= 8 )
Solution: According to above formula 8 x 10 x  17/8 = 170

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=4
Apply the formula [n(n+1)(2n+1)/6]
= 4*5*9/6
Answer : 30

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 dimension
Here n = 4, m = 3
Answer : 20

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

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 grid
Put n = 8 in the above formula
Answer : 1296