Counting figures sometimes sound too difficult if you don’t have an idea of how to solve them. In this article, we will be talking about tricks for counting figures problem.

These are most repeatedly asked reasoning questions. In this, a shape or a figure would be given to you and your task is to determine the shape and count it down within that particular shape.
For example, consider the below figure:

They simply ask in the question, how many triangles are visible in the given figure? Initially, it might look easy but it is obviously not! Without knowing tricks you can’t easily figure out correct answer.

Types of questions with Triangle Counting Problem Examples

• When a triangle  is divided  by  vertical lines

  • In such a case, we use can n (n+1)/2 where n is the number of triangles inside the main triangle

Q.  Find the number of triangles in the diagram

Number of  triangles  inside the  main triangle   =   4

4(4+1)/2

= 20/2    =    10

•  When a  triangle divided  by the horizontal lines

  • In such a case, count the number of horizontal lines that would give you the total number of triangles

Q.  Find the number  of figures  in the diagram

Total number of horizontal lines = 3

So total possible triangles are 3

• When a triangle has both vertical and horizontal lines

  • n(n+2)/2 (for vertical lines ) * number of horizontal lines

Q.  Find the number  of triangles in the diagram

Number of triangles from vertical lines (ignoring horizontal lines) = 3 = 3(3+1)/2 = 6

Horizontal lines  = 3

= 6* 3

= 18

• When a triangle has an inner triangle touching all of the edges of the outer one

  • In such a case, there always 4 figures are made and add 1 for the main triangle

Q.  Find the number  of triangles in the diagram

Starting from inner combination there are  = 4 triangle

outer combination there are  = 4 triangle

And for main  = 1

total = 9 triangles

• When a triangle has a triangle touching all of the edges of outer one and also has lines from each vertex of the main triangle touching edges of the inner one

  • In this first, ignore all the lines coming from corners of the triangle
  • Now the triangle you see would  be similar to the previous case triangle ( touching all the edges of outer triangle)
  • Count the number of triangles
  • After that consider lines coming from edges, each line would make 2 triangles
  • Now add them all together

Step 1.  If we ignore the lines coming from edges, the figure would look like

Step 2.  From the inner triangle, we will get 4, from outer one = 4, Main triangle = 1

Step 3. Total triangles = 9

Step 4. Now add lines which were removed previously, if each line creates 2 triangles then 3 lines would create = 3*2 =6 triangle

Step 5. Add them all. 9 + 6 = 15 triangles

•  When a triangle has multiple triangles made on horizontal lines 

  • If you look closely from every edge there would equal number of triangles
  • Count triangles from the base
  • Add them in a consecutive manner where the result of the previous one would be used as input for the next one. i.e 1 , 1 +2 = 3, 3+3=6, 4+6 =10′. Add all the sums together.
  • Now after adding them all up, also consider second last sum again
  • Also add it

Number triangles in each base = 3

Make a counting from 1 to 3

Add them up = 1 + 3 + 6 = 10

Now also consider second last item = 3

Total triangle 10 + 3 = 13

Counting Figures Practice Questions

1. How many triangles are there in the given figure?

2. How many triangles are there in the given figure?

3.  Find the number of triangles in the given figure :

Frequently Asked Questions:

What is the Figure Counting Reasoning?

It comes under non verbal reasoning section where one requires to identify and count the exact number of shapes present in the given figure or design.