Counting of Figures

Counting of Figures
(1) Counting of Triangles:-

In the counting of triangles, we have a figure. From the given figure we will have to

 identify how many triangles in a given known figure . 

 TYPE 1:-
Example 1. consider the following figure

                     There are 3 triangles in figure.

                     In single horizontal line sum of inside triangles 1+2=3

Example 2. consider the following figure

                    There are 6 triangles in figure.1,2,3,4,1+2,2+3

                    In single horizontal line sum of inside triangles 1+2+3=6

Example 3. consider the following figure

      There are 10 triangles in figure.1,2,3,4,1+2,2+3,3+4,1+2+3,2+3+4,(1+2+3+4) greater triangle

                    In single horizontal line sum of inside triangles 1+2+3+4=10

 TYPE 2:-

Example 4. consider the following figure

There are 6 triangles in figure.1,2,1+2,1+3,2+4,(1+2+3+4) greater triangle

                    In two horizontal lines sum of inside triangles 2(1+2)=6.

Example 5. consider the following figure

There are 20 triangles in figure.2✖{1,2,3,4,1+2,2+3,3+4,1+2+3,2+3+4,(1+2+3+4) greater triangle} 

                    In two horizontal lines sum of inside triangles 2(1+2+3+4)=20.

Example 6. consider the following figure

There are 30 triangles in figure.3✖{1,2,3,4,1+2,2+3,3+4,1+2+3,2+3+4,(1+2+3+4) greater triangle} 

                    In three horizontal lines sum of inside triangles 3(1+2+3+4)=30.

 TYPE 3:-

Example 7. consider the following figure

There are 5 triangles in figure.

Triangle inside triangle 1 hence number of triangles = 4✖1 +1 (greater triangle) = 5 

Example 8. consider the following figure

There are 9 triangles in figure.

Triangle inside triangles 2 hence number of triangles = 4✖2 +1 (greater triangle) = 9 

Example 9. consider the following figure

There are 13 triangles in figure.

Triangle inside triangles 3 hence number of triangles = 4✖3 +1 (greater triangle) = 9 

TYPE 4:-

Example 10. consider the following figure

There are 11 triangles in figure.

Triangle inside triangle 1 hence number of triangles = 4✖1 +1 (greater triangle) = 5 

and triangles 6 are (1,2,3,4,5,6) 

Hence total triangles 5+6 = 11

Example 11. consider the following figure

There are 15 triangles in figure.

Triangle inside triangle 2 hence number of triangles = 4✖2 +1 (greater triangle) = 9 

and triangles 6 are (1,2,3,4,5,6) 

Hence total triangles 9+6 = 15

TYPE 5:-

Example 12. consider the following figure

Triangles 1,2,3,4,1+2,2+3,3+4,4+1

i.e. 4✖2=8

There are 8 triangles

Example 13. consider the following figure

Example 14. consider the following figure

Example 15. consider the following figure

Example 16. consider the following figure

Example 17. consider the following figure

Example 18. consider the following figure

Example 19. consider the following figure

Example 20. consider the following figure

(2) Counting of Square:-

In the counting of squares, we have a figure. From the given figure we will have to

 identify how many squares in a given known figure . 
Type-1
If number of rows = number of columns =n
Then number of squares = ∑n²

Example 1.:-consider the following figure

Type-2
If number of rows = n,number of columns =m

Then number of squares = m✖n +(m-1)✖(n-1) + ............

Example 2.:-consider the following figure

Type-3

Example 3. :-consider the following figure