**Counting of Triangles in the Figure**

**How Many Triangles are there in the Given Figure?**

**>>Figure Type1:**Triangle having number of bisects with vertex

**>>**

**Figure Type**

**2:**Square, Rectangle and Quadrilateral

**>>**

**Figure Type**

**3:**Particular Pattern of Triangle

**>>**

**Figure Type**

**4:**Embedded Triangle having number of bisects with vertex

**>>**

**Figure Type**

**5:**Triangle having number of bisects with vertex and Horizontal Lines

**>>Figure Type1: Triangle having number of bisects with vertex**

**Q1.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of blocks made by bisects with vertex and add the number to know total number of triangles in this type of figures]

Number of triangles=1+2=3

**Q2.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of blocks made by bisects with vertex and add the number to know total number of triangles in this type of figures]

Number of triangles=1+2+3=6

**Q3.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of blocks made by bisects with vertex and add the number to know total number of triangles in this type of figures]

Number of triangles=1+2+3+4=10

**>>Figure Type2: Square, Rectangle and Quadrilateral**

**Q4.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of blocks and multiply it with number of diagonals drawn to know total number of triangles in this type of figures]

Number of triangles=4*2=8

**Q5.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of blocks and multiply it with number of diagonals drawn to know total number of triangles in this type of figures]

Number of triangles=8*2=16

**Q6.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**do separate operation of each square first of Counting the number of blocks and multiply it with number of diagonals drawn and then combine both square and then see the triangle developed in shaded region and then add all]

Number of triangles=4*2 + 4*2 + 2= 8+8+2=18

**Q7.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**do separate operation of each square first of Counting the number of blocks and multiply it with number of diagonals drawn and then combine both square and then see the triangle developed in shaded region and then add all]

Number of triangles=4*2+4*2+4*2+2+2=8+8+8+4=28

**>>Figure Type3: Particular Pattern of Triangle**

**Q8.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of blocks made by vertical line(x) and number of blocks made by horizontal line(y) and apply the formula given below to know total number of triangles in this type of figures]

Number of triangles=[x*y(x+1)]/2

Number of blocks made by vertical line(x)=1=ABC

and Number of blocks made by horizontal line(y)=2=BCED and ADE

Therefore,

Number of triangles=[1*2(1+1)]/2

=[2(2)]/2

=[4]/2

=2

**See & Throw Solution-**Number of triangles=2=ABC and ADE

**Q9.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of blocks made by vertical line(x) and number of blocks made by horizontal line(y) and apply the formula given below to know total number of triangles in this type of figures]

Number of triangles=[x*y(x+1)]/2

In the figure below,

Number of blocks made by vertical line(x)=2=ABF and AFC

and Number of blocks made by horizontal line(y)=2=BCED and ADE

Therefore,

Number of triangles=[x*y(x+1)]/2

=[2*2(2+1)]/2

=[4(3)]/2

=12/2

=6

**Q10.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of blocks made by vertical line(x) and number of blocks made by horizontal line(y) and apply the formula given below to know total number of triangles in this type of figures]

Number of triangles=[x*y(x+1)]/2

In the figure below,

Number of blocks made by vertical line(x)=4=ABC,ACD,ADE and AEF

and Number of blocks made by horizontal line(y)=3=HBFJ, GHJI and AGI

Therefore,

Number of triangles=[x*y(x+1)]/2

=[4*3(4+1)]/2

=[12(5)]/2

=60/2

=30

**Q11.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of blocks made by vertical line(x) and number of blocks made by horizontal line(y) and apply the formula given below to know total number of triangles in this type of figures]

Number of triangles=[x*y(x+1)]/2

In the figure below,

Number of blocks made by vertical line(x)=5=ABC, ACD, ADE, AEF and AFG

and Number of blocks made by horizontal line(y)=3=IBGK,HIKJ and AHJ

Therefore,

Number of triangles=[x*y(x+1)]/2

=[5*3(5+1)]/2

=[15(6)]/2

=90/2

=45

**>>Figure Type4: Embedded Triangle having number of bisects with vertex**

**Q12.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of triangles embedded inside of the triangle and count horizontal blocks and put it in the formula given below to know total number of triangles in this type of figures]

Number of triangles=4n+m

where n= number of triangles embedded inside of the triangle

and m= number of horizontal blocks

In the figure below,

Number of triangles embedded inside of the triangle, n = 1(ECF)

Number of horizontal blocks, m=1(BDFE)

Number of triangles=4n+m

=4*1+1

=4+1

=5

**Q13.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of triangles embedded inside of the triangle and count horizontal blocks and put it in the formula given below to know total number of triangles in this type of figures]

Number of triangles=4n+m

where n= number of triangles embedded inside of the triangle

and m= number of horizontal blocks

In the figure below,

Number of triangles embedded inside of the triangle, n = 2(BEC and GHI)

Number of horizontal blocks, m=1(BDFC)

Number of triangles=4n+m

=4*2+1

=8+1

=9

**Q14.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of triangles embedded inside of the triangle and count horizontal blocks and put it in the formula given below to know total number of triangles in this type of figures]

Number of triangles=4n+m

where n= number of triangles embedded inside of the triangle

and m= number of horizontal blocks

In the figure below,

Number of triangles embedded inside of the triangle, n = 3(ECF, GHI and JLK)

Number of horizontal blocks, m=1(BDFE)

Number of triangles=4n+m

=4*3+1

=12+1

=13

**>>Figure Type5: Triangle having number of bisects with vertex and Horizontal Lines**

**Q15.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of unit triagnles in a side and put it in the formula given below to know total number of triangles in this type of figures]

Number of triangles=[x(x+2)(2x+1)]/8

[Note- Take integer part only if you get decimal digits in the operation.]

In the figure below,

Number of unit triagnles in a side=3

Number of triangles=[x(x+2)(2x+1)]/8

=[3(3+2)(2*3+1)]/8

=[3*5(6+1)]/8

=[3*5*7]/8

=105/8

=13.125=13

**Q16.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of unit triagnles in a side and put it in the formula given below to know total number of triangles in this type of figures]

Number of triangles=[x(x+2)(2x+1)]/8

[Note- Take integer part only if you get decimal digits in the operation.]

In the figure below,

Number of unit triagnles in a side=4

Number of triangles=[x(x+2)(2x+1)]/8

=[4(4+2)(2*4+1)]/8

=[4(6)(8+1)]/8

=[4*6*9]/8

=216/8

=27

**Q17.How Many Triangles are there in the Given Figure?**

**Ans & Solution-**

**[Shortcut-**Count the number of unit triagnles in a side and put it in the formula given below to know total number of triangles in this type of figures]

Number of triangles=[x(x+2)(2x+1)]/8

[Note- Take integer part only if you get decimal digits in the operation.]

In the figure below,

Number of unit triagnles in a side=13

Number of triangles=[x(x+2)(2x+1)]/8

=[13(13+2)(2*13+1)]/8

=[13(15)(26+1)]/8

=[13*15*27]/8

=5265/8

=658.125=658

**Answer the following Questions made for you, write your answer in the comment box-**

**Q1. How Many Triangles are there in the Given Figure?**

**Q2. How Many Triangles are there in the Given Figure?**

**Q3. How Many Triangles are there in the Given Figure?**

**Q4. How Many Triangles are there in the Given Figure?**

**Q5. How Many Triangles are there in the Given Figure?**

**Q6. How Many Triangles are there in the Given Figure?**

**Q7. How Many Triangles are there in the Given Figure?**

**Q8. How Many Triangles are there in the Given Figure?**

**Q9. How Many Triangles are there in the Given Figure?**

**Q10. How Many Triangles are there in the Given Figure?**

**Q11. How Many Triangles are there in the Given Figure?**

**Q12. How Many Triangles are there in the Given Figure?**

**Q13. How Many Triangles are there in the Given Figure?**

**Q14. How Many Triangles are there in the Given Figure?**

**Q15. How Many Triangles are there in the Given Figure?**

**Q16. How Many Triangles are there in the Given Figure?**

**Q17. How Many Triangles are there in the Given Figure?**

**Q18. How Many Triangles are there in the Given Figure?**

**Q19. How Many Triangles are there in the Given Figure?**

**Q20. How Many Triangles are there in the Given Figure?**

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