Counting of Figures - Expert Level
Practice and master this topic with our carefully crafted questions.
Challenge yourself with expert-level counting of figures questions featuring complex, overlapping and mixed shapes. Improve observation, accuracy and problem-solving speed.
The maximum number of squares in the given figure is
š Analyze the Pattern
On carefully observing the labelled figure:
ā» Count the Squares
The squares can be counted according to their sizes:
- 1 Ć 1 Squares (9): AGPF, GHOP, HBIO, FENP, HIMO, ENLM, MJKI, LKCM, EDCJ
- 2 Ć 2 Squares (4): AHNE, GBJO, FLKD, PICJ
- 3 Ć 3 Square (1): ABCD
Total Squares = 9 + 4 + 1 = 14
š Quick Formula Method
For an 3 Ć 3 square grid, the total number of squares is calculated as:
1Ā² + 2Ā² + 3Ā²
= 9 + 4 + 1
= 14 Squares
General Formula:
1Ā² + 2Ā² + 3Ā² + ... + nĀ²
where n is the number of small squares on one side.
Correct Answer: Option (D) ā 14 Squares
How many triangles are there in the figure PQRS ?
š Analyze the Pattern
On carefully observing the labelled figure:
šŗ Count the Triangles
The triangles present in the figure are:
PAE, ADE, AQE, QBE, DEC, CES, BER, ECR, PDE, EQD, ESR, PRB, PSQ, PQR, PSR, SQR
Total Triangles = 16
Correct Answer: Option (A) ā 16 Triangles
How many squares are there in the square figure ABCD ?
š Analyze the Pattern
On carefully observing the labelled figure:
ā» Count the Squares
The squares present in the figure are:
- 1 Ć 1 Squares (16): AQOP, ERUQ, FSVR, GBHS, TOWN, UXWT, VYXI, DMXW, QRET, RSVU, STVH, WXLM, XYKL, OPND, HIJC, ICKJ
- 2 Ć 2 Squares (9): AEUO, EFVR, FGHS, PWTD, RXYL, SVIK, ARTD, EXVL, FSKC
- 3 Ć 3 Squares (4): AFXO, EGYL, PBID, RHJC
- 4 Ć 4 Square (1): ABCD
Total Squares = 30
š Quick Formula Method
For a 4 Ć 4 square grid:
1Ā² + 2Ā² + 3Ā² + 4Ā²
= 16 + 9 + 4 + 1
= 30 Squares
where n = 4 (number of small squares on one side)
Correct Answer: Option (4) ā 30 Squares
How many triangles are there in the given figure ?
š Analyze the Pattern
On carefully observing the labelled figure:
šŗ Count the Triangles
The triangles present in the figure are:
EIB, EIA, AIM, ABM, BIM, EAB, HLA, HLD, LDA, ADM, DKM, DKC, KCM, DCM, GKC, GKB, BJF, BJC, CJF, CFM, BCF, HDA, HDB, HFG, EFH, EFG, GHF, EHG
Total Triangles = 28
Correct Answer: Option (B) ā 28 Triangles
How many triangles are there in the following figure ?
š Analyze the Pattern
On carefully observing the labelled figure:
šŗ Count the Triangles
The triangles present in the figure are:
šŗ ABC; šŗ ABE; šŗ ACE; šŗ ABD; šŗ ACF; šŗ ABD; šŗ BCD; šŗ AFG; šŗ ADG; šŗ BFG; šŗ BEG; šŗ CEG; šŗ CDG; šŗ ACG; šŗ ABG; šŗ BCG
Total Triangles = 16
Correct Answer: Option (A) ā 16 Triangles
How many rectangles are there in the given figure ?
š Analyze the Pattern
On carefully observing the labelled figure:
ā Count the Rectangles
The rectangles present in the figure are:
AEML, LMOK, AEOK, KNJD, NOIJ, KOID, AEID, LMID, EFPO, FBGP. EBGO, OGHQ, QHCI, OGCI, EBHQ, EBCI, AFPK, ABGK, NGCJ, KGCD, ABCD.
Total Rectangles = 21
Correct Answer: Option (C) ā 21 Rectangles
How many rectangles are there in the figure given ?
š Analyze the Pattern
On carefully observing the labelled figure:
ā Count the Rectangles
The rectangles present in the figure are:
ABCD, AEJK, AFIK, ABGK, EFIJ, EBGJ, FBGI, FBCH, IGCH, AFHD, KIHD
Total Rectangles = 11
Correct Answer: Option (4) ā 11 Rectangles
How many triangles are there in the following figure ?
š Analyze the Pattern
On carefully observing the labelled figure:
šŗ Count the Triangles
The triangles present in the figure are:
šŗABC ; šŗABD ; šŗADC ; šŗAFC; šŗFDC ; šŗAFB ; šŗFDB ; šŗFBC; šŗGBD ; šŗADE ; šŗGBE ; šŗFDG; šŗDBE ;
Total Triangles = 13
Correct Answer: Option (B) ā 13 Triangles
How many triangles are there in the following figure ?
š Solution
After carefully observing the labelled figure, count the triangles from the smallest to the largest.
šŗ Small Triangles (16)
AGH, AHB, BHI, BIC, GHI, HIB, EGJ, GJE, JID, IDC, GJI, JID, FAG, FGE, BCI, ICD
šŗ Medium Triangles (8)
ABI, AGI, EGD, GCD, FAE, BCD, FGI, GIC
šŗ Large Triangles (4)
FAB, ABC, FED, EDC
Total Triangles = 16 + 8 + 4 = 28
ā Correct Answer: Option (C) ā 28 Triangles
How many triangles are there in the following figures?
š Solution
After carefully observing the labelled figure, count the triangles from the smallest to the largest.
šŗ Small Triangles (16)
ADE, DEG, DGE, FJG, FGJ, EGH, GHK, GKL, IJM, JMN, JNK, KNO, KOL, LOC, BIM, HOC
šŗ Medium Triangles (7)
DFE, FGH, IJK, JKL, FKN, GKO, IMN
šŗ Large Triangles (3)
AFH, DLC, BEC
šŗ Largest Triangle (1)
ABC
Total Triangles = 16 + 7 + 3 + 1 = 27
ā Correct Answer: Option (B) ā 27 Triangles