Figure Counting: Meaning, Reasoning Questions with Answers, Formula

The topic of 'Figure Counting' is a part of Logical Reasoning that appears in various government exams. This topic is based on Non-Verbal Reasoning. This topic is commonly found in many Government exams as well as entrance exams which are BITSAT, SSC, Railway, Defence, and other competitive exams. The problems in this topic involve counting the number of figures of different sizes and shapes in the given adjoining figure, such as triangles, squares, rectangles, quadrilaterals, etc.

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This Story also Contains

1. Approach to Solve the Questions Based on Figure Counting Reasoning
2. Types of Questions Asked from the Figure Counting Reasoning
3. Recommended Books and Online Resources
4. Question Weightage of Figure Counting in Competitive Exams
5. Practice Questions For Counting the Squares
6. Practice Questions For Counting the Rectangles
7. Practice Questions For Counting the Triangles
8. Practice Questions For Counting the Quadrilaterals
9. Some Figure Counting Tricks to Solve the Questions of Figure Counting
10. Figure Counting Questions for BITSAT/ CUET
11. Figure Counting Questions for SSC CGL/ SSC CHSL/ SSC CPO/ SSC Stenographer exams
12. Figure Counting Questions for Railway Recruitment Board Exams

Approach to Solve the Questions Based on Figure Counting Reasoning

To solve questions related to figure counting reasoning, we must first identify the figure to count. The best approach is to start the counting of figures from the bigger ones. For instance, let’s assume that a counting figure question asks to find the number of triangles in the given figure. For this, we have to label the figure and then point out the different vertices that together make the complete triangles. From this, we will be able to identify the different triangles and track the number of triangles in the figure. The same needs to be followed for other shapes as well.

Types of Questions Asked from the Figure Counting Reasoning

The topic of figure counting reasoning is divided into various parts based on the counting of different shapes and they are as follows -

1. Count the total number of Squares

2. Count the total number of rectangles

3. Count the total number of triangles

4. Count the total number of quadrilaterals

Let’s discuss these types and figure counting formula in detail with the help of examples -

1. Count the Total Number of Squares

In this type, we have to count the total number of squares in the given adjoining figure. For this, label the given adjoining figure, and point out the different vertices that together make the complete square. By using these vertices, we can count the number of squares in the given figure. Let’s understand this with the help of an example.

Example: Consider the diagram given below and count the number of squares.

To count the number of squares in the above adjoining figure, first, we have to label the figure and then count the number of squares. The given figure can be labelled as shown below:

The squares in the above figure are - QYUW, WUXT, YRVU, UVSX, QRST, IJUP, PUNO, ULMN, JKLU, IKMO, GFUH, HUBA, UDCB, FEDU, and GECA. Thus, there are a total of 15 squares in the above figure.

2. Count the Total Number of Rectangles

In this type, we have to count the total number of rectangles in the given adjoining figure. For this, label the given adjoining figure, and point out the different vertices that together make the complete rectangle. By using these vertices, we can count the number of rectangles in the given figure. However, while counting the number of rectangles, we have to consider the squares as well. All squares can be rectangles but all rectangles cannot be squares. Therefore, in the counting of rectangles, we have to count the number of squares in the given figure. Let’s understand this with the help of an example.

Example: Consider the diagram given below and count the number of rectangles.

To count the number of rectangles in the above adjoining figure, first, we have to label the figure and then count the number of rectangles. The given figure can be labelled as shown below:

The rectangles in the above figure are - ABCD, EFGH, IJKL, MFNQ, EMQR, EFNR, QNOD, NJPO, QJPD, RNGH, MFOD. Thus, there are a total of 11 rectangles in the above figure.

3. Count the Total Number of Triangles

In this type, we have to count the total number of triangles in the given adjoining figure. For this, label the given adjoining figure, and point out the different vertices that together make the complete triangle. By using these vertices, we can count the number of triangles in the given figure. Let’s understand this with the help of an example.

Example: Consider the diagram given below and count the number of triangles.

To count the number of triangles in the above adjoining figure, first, we have to label the figure and then count the number of triangles. The given figure can be labelled as shown below:

The triangles in the above figure are - AFE, EDG, EFG, EHN, HNM, EHM, HIJ, GML, GNK, and HKL. Thus, there are a total of 10 triangles in the above figure.

4. Count the Total Number of Quadrilaterals

In this type, we have to count the total number of Quadrilaterals in the given adjoining figure. For this, label the given adjoining figure, and point out the different vertices that together make the complete quadrilaterals. By using these vertices, we can count the number of quadrilaterals in the given figure. However, while counting the number of quadrilaterals, we have to consider the squares and rectangles as well. All the four-sided enclosed figures are quadrilaterals, irrespective of the shape. Let’s understand this with the help of an example.

Example: Consider the diagram given below and count the number of quadrilaterals.

To count the number of quadrilaterals in the above adjoining figure, first, we have to label the figure and then count the number of quadrilaterals. The given figure can be labelled as shown below:

The quadrilaterals in the above figure are - ABFE and ABCE. Thus, there are a total of 2 quadrilaterals in the above figure.

The important non verbal reasoning topics are given below:

Recommended Books and Online Resources

1. A Modern Approach to Verbal & Non-Verbal Reasoning by R.S. Aggarwal

2. SSC Reasoning by Rakesh Yadav

3. For practice you should solve the various figure counting reasoning questions pdf available online.

Question Weightage of Figure Counting in Competitive Exams

The number of questions based on figure counting varies from exam to exam -
1) Questions asked in SSC exams i.e. SSC MTS, SSC CGL, SSC CHSL, SSC CPO, Steno - 2 to 3 questions.
2) Questions asked in the RRB exams i.e. Group D, NTPC, JE, ALP etc - 2 to 3 questions.
3) Questions asked in Banking exams, Bank PO, Bank Clerk - 1 to 2 questions.

For practice follow the below figure counting questions with answers of each type:

Practice Questions For Counting the Squares

Q1. Directions: How many squares are there in the given figure?

A) 10

B) 11

C) 12

D) 14 (Correct)

Solution: Label the given figure and then count the number of squares. The given figure can be labelled as shown below:

The squares in the above figure are - ABCP, PCQO, OQRN, NRLM, CDEQ, QESR, RSKL, EFGS, SGJK, GHIJ, PDSN, OEKM, QFJL and NLSQ. Thus, there are 14 squares in the above figure. Hence, the fourth option is correct.

Q2. Directions: How many squares are there in the given figure?

A) 8

B) 14

C) 10 (Correct)

D) 12

Solution: Label the given figure and then count the number of squares. The given figure can be labelled as shown below:

The squares in the above figure are - ABIH, HIFG, BCDI, IDEF, JKLM, JOIN, OKPI, IPLQ, NIQM, and ACEG. Thus, there are 10 squares in the above figure. Hence, the third option is correct.

Q3. Directions: How many squares are there in the given figure?

A) 28 (Correct)

B) 54

C) 53

D) 27

Solution: Label the given figure and then count the number of squares. The given figure can be labelled as shown below:

The squares in the above figure are - ABIH, HION, BCJI, OIJP, ACPN, DKLE, ELMF, KLRQ, LRSM, DQSF, FSTG, QSZY, ZSTa, YDGa, eUVf, fbcr, rcdg, VbcW, WcdX, fVXg, lefm, mfgn, hiks, iktj, oskp, pktq, ohjq, and lUXn. Thus, there are 28 squares in the above figure. Hence, the first option is correct.

Q4. Directions: How many squares are there in the given figure?

A) 26

B) 30

C) 35 (Correct)

D) 38

Solution: Label the given figure and then count the number of squares. The given figure can be labelled as shown below:

The squares in the above figure are - ABQP, BCXQ, CXRD, DRFE, POSQ, SQXT, TXRU, RUGF, OSVN, VSTY, YTUW, WUGH, NMLV, LKYV, KYWJ, JIHW, ACTO, BDUS, CTGE, NPXY, VQRW, YXFH, MOTK, LSUJ, KTGI, NADW, VBEH, MPRJ, LQFI, AEIM, KOCG, VOQT, TQCR, TRGW, and KVTW. Thus, there are 35 squares in the above figure. Hence, the third option is correct.

Q5. Directions: How many squares are there in the given figure?

A) 11

B) 12

C) 10 (Correct)

D) 9

Solution: Label the given figure and then count the number of squares. The given figure can be labelled as shown below:

The squares in the above figure are - LMNO, EFGH, EJIT, TIKH, JFUI, IUGK, LPIR, RIQO, IPMS, and ISNQ. Thus, there are 10 squares in the above figure. Hence, the third option is correct.

Practice Questions For Counting the Rectangles

Q1. Directions: How many rectangles are there in the given figure?

A) 8

B) 5

C) 9 (Correct)

D) 4

Solution: Label the given figure and then count the number of rectangles. The given figure can be labelled as shown below:

The rectangles in the above figure are - AEOH, EBFO, OFCG, HOGD, ABFH, HFCD, AEGD, EBCG, and ABCD. Thus, there are 9 rectangles in the above figure. Hence, the third option is correct.

Q2. Directions: How many rectangles are there in the figure ABCD?

A) 11

B) 12

C) 9

D) 10 (Correct)

Solution: Label the given figure and then count the number of rectangles. The given figure can be labelled as shown below:

The rectangles in the above figure are - AEJF, EBIJ, KICH, NGDH, KLMN, FGML, FGNK, ABCD, AFIB, and FDCI. Thus, there are 10 rectangles in the above figure. Hence, the fourth option is correct.

Q3. Directions: How many rectangles are there in the given figure?

A) 8

B) 5

C) 6 (Correct)

D) 4

Solution: Label the given figure and then count the number of rectangles. The given figure can be labelled as shown below:

The rectangles in the above figure are - ABDC, CDFE, ABFE, ONGF, MLJK, and GKIH. Thus, there are 6 rectangles in the above figure. Hence, the third option is correct.

Q4. Directions: How many rectangles are there in the given figure?

A) 9 (Correct)

B) 7

C) 8

D) 10

Solution: Label the given figure and then count the number of rectangles. The given figure can be labelled as shown below:

The rectangles in the above figure are - ABCF, CDEF, ABDE, MNGL, LGHK, KHIJ, MNHK, MNIJ, and LGIJ. Thus, there are 9 rectangles in the above figure. Hence, the first option is correct.

Q5. Directions: How many rectangles are there in the given figure?

A) 6

B) 7

C) 8

D) 9 (Correct)

Solution: Label the given figure and then count the number of rectangles. The given figure can be labelled as shown below:

The rectangles in the above figure are - ABJI, BJLC, CDEK, EFGK, GHIL, ACGH, AHFD, AILC, and CDFG. Thus, there are 9 rectangles in the above figure. Hence, the fourth option is correct.

Practice Questions For Counting the Triangles

Q1. Directions: How many triangles are there in the given figure?

A) 24

B) 30

C) 28 (Correct)

D) 29

Solution: To solve the figure counting triangle questions label the given figure and then count the number of triangles. The given figure can be labelled as shown below:

The triangles in the above figure are - AFG, AEG, AFE, BFJ, BDJ, BFD, CEI, CDI, CED, ABD, ACD, ABC, AHB, AHE, DEF, DEG, DFG, GHE, DEH, JHD, BEC, BHD, AEB, AFD, AED, DJE, FEJ, and BDE. Thus, there are 28 triangles in the above figure. Hence, the third option is correct.

Q2. Directions: How many triangles are there in the given figure?

A) 8

B) 10

C) 12 (Correct)

D) 14

Solution: To solve the figure counting triangle questions label the given figure and then count the number of triangles. The given figure can be labelled as shown below:

The triangles in the above figure are - ADE, BCF, AJD, DJC, CJB, BJA, ADC, DCB, CBA, BAD, EAC, and DBF. Thus, there are 12 triangles in the above figure. Hence, the third option is correct.

Q3. Directions: How many triangles are there in the given figure?

A) 4

B) 5

C) 7

D) 6 (Correct)

Solution: To solve the figure counting triangle questions label the given figure and then count the number of triangles. The given figure can be labelled as shown below:

The triangles in the above figure are - ABC, DEF, GHI, JKC, MNC, and LOF. Thus, there are 6 triangles in the above figure. Hence, the fourth option is correct.

Q4. Directions: How many triangles are there in the given figure?

A) 9

B) 10

C) 11

D) 8 (Correct)

Solution: To solve the figure counting triangle questions label the given figure and then count the number of triangles. The given figure can be labelled as shown below:

The triangles in the above figure are - AEF, GBH, ICJ, LDK, MNO, ONQ, MOQ, and MPQ. Thus, there are 8 triangles in the above figure. Hence, the fourth option is correct.

Q5. Directions: How many triangles are there in the given figure?

A) 20

B) 21

C) 22

D) 24 (Correct)

Solution: To solve the figure counting triangle questions label the given figure and then count the number of triangles. The given figure can be labelled as shown below:

The triangles in the above figure are - ABJ, AIJ, IJK, BJK, ILF, FLE, LEK, ILK, BCK, CKD, KDE, AHI, HIG, GIF, AIK, BIK, IEF, KFE, IKF, IKE, ABK, AIB, AKF, and BIE. Thus, there are 24 triangles in the above figure. Hence, the fourth option is correct.

Practice Questions For Counting the Quadrilaterals

Q1. Directions: How many quadrilaterals are there in the given figure?

A) 17 (Correct)

B) 15

C) 14

D) 21

Solution: Label the given figure and then count the number of quadrilaterals. The given figure can be labelled as shown below:

The quadrilaterals in the above figure are - AFKE, FBGK, EKHD, KGCH, ABGE, EGCD, AFHD, FBCH, ABCD, AFJD, FBCJ, EKJD, KGCJ, ABIE, FBIK, EICD, and KICH. Thus, there are 17 quadrilaterals in the above figure. Hence, the first option is correct.

Q2. Directions: How many quadrilaterals are there in the given figure?

A) 7

B) 6 (Correct)

C) 4

D) 8

Solution: Label the given figure and then count the number of quadrilaterals. The given figure can be labelled as shown below:

The quadrilaterals in the above figure are - ADFE, FEBC, ADCE, BCFA, AFCE, and ABCD. Thus, there are 6 quadrilaterals in the above figure. Hence, the second option is correct.

Q3. Directions: How many quadrilaterals are there in the given figure?

A) 16 (Correct)

B) 17

C) 15

D) 14

Solution: Label the given figure and then count the number of quadrilaterals. The given figure can be labelled as shown below:

The quadrilaterals in the above figure are - ABOK, BCPO, CDER, ERGF, GHNQ, JNHI, JKLM, LPQM, KPQJ, IJQG, ACPK, CDFG, ADFI, ACGI, KPGI, and ACQJ. Thus, there are 16 quadrilaterals in the above figure. Hence, the first option is correct.

Q4. Directions: How many quadrilaterals are there in the given figure?

A) 15

B) 16

C) 14 (Correct)

D) 13

Solution: Label the given figure and then count the number of quadrilaterals. The given figure can be labelled as shown below:

The quadrilaterals in the above figure are - ABML, KLMN, ABNK, BCDN, ACDK, DEJK, EFPO, FGHP, EGHO, OHIJ, EGIJ, ACGI, ACEJ, and GIKD. Thus, there are 14 quadrilaterals in the above figure. Hence, the third option is correct.

Q5. Directions: How many quadrilaterals are there in the given figure?

A) 21

B) 22

C) 24 (Correct)

D) 23

Solution: Label the given figure and then count the number of quadrilaterals. The given figure can be labelled as shown below:

The quadrilaterals in the above figure are - AEFO, AEGP, AEHD, OFGP, OFHD, PGHD, ABMO, EBMF, OIJP, OMNP, FIJG, FMNG, IMNJ, PKLD, PNCD, GKLH, GNCH, KNCL, EBNG, EBCH, FMCH, ABNP, ABCD, and OMCD. Thus, there are 24 quadrilaterals in the above figure. Hence, the third option is correct.

Some Figure Counting Tricks to Solve the Questions of Figure Counting

1. Count the Number of Squares and Rectangles

To count the number of squares and rectangles in a figure, firstly, we have to number the divided parts in the figure as shown below.

To determine the number of squares, multiply the last number in the row with the last number in the column and then multiply the numbers after subtracting 1 from both numbers. Continue the procedure until one of the two numbers becomes 1 and then add all the resultant numbers to get the number of squares.

Number of Squares = (6 × 3) + (5 × 2) + (4 × 1) = 18 + 10 + 4 = 32

In the above figure, there are a total of 32 squares.

To determine the number of rectangles, multiply the sum of the numbers in the first row and the sum of the numbers in the first column.

Number of total rectangles = (1 + 2 + 3 + 4 + 5 + 6) × (1 + 2 + 3) = 21 × 6 = 126

In the above figure, there are a total of 126 rectangles. (These rectangles also include the count of squares, because all squares are rectangles but all rectangles are not squares.)

There is one more figure counting trick to calculate the number of squares and rectangles in a square (when the number of rows is equal to the number of columns)

The number of squares = [n(n + 1)(2n + 1)]/6

The number of rectangles = [n(n + 1)/2]²

Here, n = 6

So, the total number of squares = (6 × 7 × 13)/6 = 546/6 = 91

The total number of rectangles = [(6 × 7)/2]² = [42/2]² = [21]² = 441 (Squares are also included)

NOTE: But, in case, if the question asks “only rectangles” instead of “total rectangles”, then subtract the number of squares from the total number of rectangles.

2. Count the Number of Triangles

Let’s consider some figures of the triangle of a common type.

In these types of triangles, we will count the total number of triangles on the basis of the number of bases in each triangle. In the first figure, we have 2 bases, in the second figure, we have 3 bases, and in the third figure, we have 4 bases. Let us consider that ‘n’ denotes the number of bases.

So, the total number of triangles = [n(n + 1)]/2

The number of triangles in the first figure = [2(2 + 1)]/2 = [6]/2 = 3

The number of triangles in the first figure = [3(3 + 1)]/2 = [12]/2 = 6

The number of triangles in the first figure = [4(4 + 1)]/2 = [20]/2 = 10

Here, the triangles can also be divided horizontally as shown below:

Here, the first step is the same as we did before but after that step here we will multiply that by the number of horizontal parts.

As we can see, in the first figure, there are 2 horizontal parts, and in the second figure, there are 3 horizontal parts.

So, the number of triangles in the first figure = 2 × [(2(2 + 1))/2] = 2 × [(6)/2] = 2 × 3 = 6

So, the number of triangles in the second figure = 3 × [(4(4 + 1))/2] = 3 × [(20)/2] = 3 × 10 = 30

Now, let’s consider the triangles of another form.

As, in the above figures, we can see a big triangle is divided into a number of small triangles. To count the number of triangles in the above figures, we have to determine the number of bases in the given figure and then apply the required formula. Let us consider that ‘n’ denotes the number of bases.

In the first figure, there are 2 bases, in the second figure, there are 3 figures, in the third figure, there are 4 bases, and in the fifth figure, there are 5 bases. So, the number of bases is even as well as odd. Based, on the number of bases (either odd or even), there are 2 formulas.

So, the total number of triangles (when the number of bases is even) = [n(n + 2)(2n + 1)]/8

So, the total number of triangles (when the number of bases is odd) = [{n(n + 2)(2n + 1)} - 1]/8

For the first figure, n = 2 (Even)

So, the total number of triangles = [2(2 + 2)(2(2) + 1)]/8 = (2 × 4 × 5)/8 = 5

For the second figure, n = 3 (Odd)

So, the total number of triangles = [{3(3 + 2)(2(3) + 1)} - 1]/8 = [(3 × 5 × 7) - 1]/8 = 104/8 = 13

For the third figure, n = 4 (Even)

So, the total number of triangles = [4(4 + 2)(2(4) + 1)]/8 = (4 × 6 × 9)/8 = 27

For the fourth figure, n = 5 (Odd)

So, the total number of triangles = [{5(5 + 2)(2(5) + 1)} - 1]/8 = [(5 × 7 × 11) - 1]/8 = 384/8 = 48

3. Count the Number of Triangles in a Square or Rectangle

If we have to count the number of triangles in a square or a rectangle, then just simply multiply the total number of parts in the figure by 2 to get the required answer. The example is as follows:

Here, in the above figure, there are a total of 8 parts of a square.

So, the total number of triangles = 2 × 8 = 16

For more questions you must practice from figure counting questions pdf or download the e-book of figure counting questions with answers pdf below:
Figure Counting

Figure Counting Questions for BITSAT/ CUET

1) Directions: How many rectangles are there in the given figure?

1) 9

2) 10

3) 11

4) 12

Solution

The given figure can be labeled as shown below –

There are a total of 11 rectangles in the above figure. They are ABCD, EFGH, IJKL, MFNQ, EMQR, EFNR, QNOD, NJPO, QJPD, RNGH, MFOD.

Hence, the third option is correct.

Figure Counting Questions for SSC CGL/ SSC CHSL/ SSC CPO/ SSC Stenographer exams

1) Directions: How many triangles are there in the given figure?

1) 11

2) 9

3) 10

4) 12

Solution

The given figure can be labeled as shown below –

There are a total of 10 triangles in the above figure. They are AFE, EDG, EFG, EHN, HNM, EHM, HIJ, GML, GNK, HKL.

Hence, the third option is correct.

2) Directions: How many triangles are there in the given figure?

1) 28

2) 36

3) 40

4) 48

Solution

The figure can be labeled as shown below –

There are a total of 36 triangles in the above figure. They are AEI, IEB, BJF, CJF, DKG, CKG, HAL, DHL, AEB, BFC, CGD, DHA, AML, LMD, DMK, KMC, CMJ, JMB, BMI, IMA, AMD, DMC, CMB, BMA, ADC, DCB, CBA, BAD, HAM, HDM, MBF, MCF, MDG, MCG, MAE, MBE.

Hence, the second option is correct.

Figure Counting Questions for Railway Recruitment Board Exams

1) Directions: How many triangles are there in the given figure?

1) 10

2) 11

3) 12

4) 13

Solution

The figure can be labeled as shown below –

There are 11 triangles in the above figure. They are ABC, AGB, AGC, DBF, DFG, DBG, GEH, HEC, GEC, ADG, AEG.

Hence, the second option is correct.

Read more: The verbal reasoning topics given below.

About the Faculty
Tanu Gupta, with over a decade of experience as a reasoning faculty, specializes in preparing students for various entrance examinations and career development. Her extensive work with multiple educational platforms and institutions has honed her expertise in logical and analytical thinking. Her dedication to innovative teaching methods ensures these articles provide practical insights and expert guidance.