Counting of Figures - Basic Level

🔍 Analyze the Pattern

On carefully observing the labelled figure:

🔺 Count the Triangles

Using the labelled figure, the triangles can be counted systematically as follows:

AEI, EOI, OHI, HAI, EBJ, BFJ, 1 FOJ, OEJ, HOL, OGL, GDL, DHL, OFK, FCK, CGK, GOK, HAE, AEO, EOH, OHA, OEB, EBF, BFO, FOE, DHO, HOG, OGD, GDH, GOF, OFC, FCG, CGO, HEF, EFG, FGH, GHE, ABO, BCO, CDO, DAO, DAB, ABC, BCD, CDA

Total Triangles = 44

◻ Count the Squares

The squares present in the figure are:

• HIOL
• IEJO
• JFKO
• KGLO
• AEOH
• EBFO
• OFGC
• HOGD
• EFGH
• ABCD

Total Squares = 10

Correct Answer: Option (A) — 44 Triangles and 10 Squares

⚡ Quick Exam Tip

For counting figures, label all intersection points first. Then:

1. Count the smallest figures.
2. Count medium-sized figures.
3. Count larger composite figures.
4. Finally, count the outermost figure.

Final Answer: 44 Triangles and 10 Squares → Option (A).

🔍 Analyze the Pattern

On carefully observing the labelled figure:

🔺 Count the Triangles

Using the labelled figure, the triangles can be counted systematically as follows:

ABI, BGI, GHI, HAI, BCJ, CFJ, FGJ, GBJ, CDK, DEK, EFK, FCK, ABG, BGH, GHA, HAB, BCF, CFG, FGB, GBC, CDE, DBF, EFC, FGD, AGC, BFD, HBF, GCE

Total Triangles = 28

◻ Count the Squares

The squares present in the figure are:

• ABGH
• BCFG
• CDEF
• ACFH
• BDEG

Total Squares = 5

Correct Answer: Option (C) — 28 Triangles and 5 Squares

🔍 Analyze the Pattern

On carefully observing the labelled figure:

🔺 Count the Triangles

Using the labelled figure, the triangles can be counted systematically as follows:

JBO, BKO, KDO, DFO, FGO, GHO, HIO, IJO,
ABJ, BCK, CKD, DBF, IBO, EDO, DGO, GIO,
ABO, CBD, DEO, IBD, BDG, DGI, GIB, ACO,
COE, ACE

Total Triangles = 26

◻ Count the Squares

The squares present in the figure are:

• BKOD
• JIOH
• OFGH
• BIOH
• DOFG
• BIGD (Outer Square)

Total Squares = 6

Correct Answer: Option (C) — 26 Triangles and 6 Squares

⚡ Quick Exam Tip

Label all vertices and intersection points first. Then:

1. Count the smallest triangles.
2. Count larger triangles formed by combining smaller ones.
3. Count the smallest squares first.
4. Finally, count the larger composite squares.

Final Answer: 26 Triangles and 6 Squares → Option (C).

🔍 Analyze the Pattern

On carefully observing the labelled figure:

🔺 Count the Triangles

Using the labelled figure, the triangles can be counted systematically as follows:

BGM, GHM, HAM, ABM,
GIN, IJN, JHN, HGN,
IKO, KLO, LJO, JIO,
KDP, DEP, ELP, LKP,
BCD, AFE,
ABG, BGH, GHA, HAB,
HGI, GIJ, IJH, JHG,
JIK, IKL, KLJ, LJI,
LKD, KDE, DEL, ELK,
BHI, GJK, ILD,
AGJ, HIL, JKE

Total Triangles = 40

◻ Count the Squares

The squares present in the figure are:

• MGNH
• NIOJ
• OKPL
• GHAB
• GIJH
• IKLJ
• KDEL

Total Squares = 7

Correct Answer: Option (C) — 40 Triangles and 7 Squares

⚡ Quick Exam Tip

To avoid missing figures, first label all vertices and intersection points. Then count:

1. The smallest triangles.
2. Composite triangles formed by combining adjacent triangles.
3. The smallest squares.
4. The larger squares formed by combining adjacent squares.

Final Answer: 40 Triangles and 7 Squares → Option (C).

🔍 Analyze the Pattern

On carefully observing the labelled figure:

🔺 Count the Triangles

Using the labelled figure, the triangles can be counted systematically as follows:

BPN, PNE, ABM, EFG, MLK, GHI, QRO,
RSO, STO, QTO, BPE, TQR, QRS, RST,
STQ, MPO, GPO, LPJ, HPJ, MPG, LPG

Total Triangles = 21

◻ Count the Squares

The squares present in the figure are:

• KJOM
• JIGQ
• ANOM
• NFGO
• CDEB
• QRST
• AFIK

Total Squares = 7

Correct Answer: Option (A) — 21 Triangles and 7 Squares

⚡ Quick Exam Tip

Label all vertices and intersection points before counting. Count the smallest figures first, then the larger composite figures to avoid missing or double-counting shapes.

1. Count the smallest triangles.
2. Count composite triangles.
3. Count the smallest squares.
4. Finally, count the larger squares formed by combining smaller ones.

Final Answer: 21 Triangles and 7 Squares → Option (A).

🔍 Analyze the Pattern

On carefully observing the figure, the centres of all the circles are joined by horizontal and vertical lines, forming a grid of squares.

◻ Count the Squares

The squares can be counted as follows:

1 × 1 Squares (6):

• ABED
• BCFE
• DEHG
• EFIH
• GHKJ
• HILK

2 × 2 Squares (2):

• ACIG
• DFLJ

Total Squares = 6 + 2 = 8

Correct Answer: Option (C) — 8 Squares

⚡ Quick Exam Tip

When counting squares in a grid:

1. Count all 1 × 1 squares first.
2. Then count all possible 2 × 2 squares.
3. Continue with larger squares, if any.

In this figure:

• 6 unit squares
• 2 composite (2 × 2) squares

Final Answer: 8 Squares → Option (C).

🔍 Analyze the Figure

To find the minimum number of straight lines required to construct the figure, count each continuous straight line only once. A line passing through one or more intersection points is still considered a single straight line as long as its direction does not change.

📏 Count the Straight Lines

The continuous straight lines in the figure are:

AE, JF, AJ, CH, EF, AG, BF, JD, IE, AB, DE, JI, FG

Total Straight Lines = 13

Correct Answer: Option (B) — 13 Straight Lines

💡 Quick Exam Tip

In questions asking for the minimum number of straight lines, count every continuous line only once. Do not split a line at intersection points unless the direction changes.

For this figure:

• Horizontal lines = 2
• Vertical lines = 3
• Diagonal lines = 8

2 + 3 + 8 = 13 Straight Lines

Final Answer: Option (B) — 13 Straight Lines.

🔍 Analyze the Figure

Count the triangles systematically by grouping them into small, medium and large triangles.

🔺 Count the Triangles

Small Triangles (10):

JHI, HFG, ACK, CHK, HJK, GEL, EFL, FHL, HCL, JAK

Medium Triangles (10):

JAC, ACH, CHJ, HJA, HCE, CEF, EFH, FHC, AHE, JCF

Large Triangles (2):

ABC, CDE

Total Triangles = 22

Correct Answer: Option (C) — 22 Triangles

💡 Quick Exam Tip

When counting triangles, begin with the smallest ones, then count all composite triangles, and finally check for the largest triangles formed by combining smaller regions.

Final Answer: 22 Triangles → Option (C).

🔍 Analyze the Figure

Count both the normal squares and the rotated (diamond-shaped) squares formed in the figure.

◻ Count the Squares

Rotated Squares (5):

• ABCK
• CDEL
• JKHI
• HLFG
• KCLH

Normal Squares (2):

• ACHJ
• CEFH

Total Squares = 7

Correct Answer: Option (C) — 7 Squares

💡 Quick Exam Tip

In counting-of-figures questions, always check for rotated (diamond) squares in addition to normal squares. The inner square KCLH is commonly overlooked, which is why many people incorrectly count only 6 squares.

Final Answer: 7 Squares → Option (C).

🔍 Analyze the Pattern

On carefully observing the labelled figure:

▱ Count the Parallelograms

The parallelograms present in the figure are:

EMLA, NIDJ, BFMG, CGNH, GMKN, FGME, GHNM, MNKL, FGNM, GHIN, MNJK, FGLA, ENKA, GHDJ, MIDK, FGJK, GHKL, FBNK, CHKM, EFHN, MFHI, FHKA, FHDK

Total Parallelograms = 23

Correct Answer: Option (A) — 23 Parallelograms