Count from Smallest Figure → Medium Figure → Largest Figure.

🔍 Count the Minimum Straight Lines

Count only the longest continuous straight lines in each direction.

Horizontal Lines

DE, FH, IL, BC = 4 Lines

Lines Parallel to AC

AC, DO, FN, IM = 4 Lines

Lines Parallel to AB

AB, EM, HN = 3 Lines

Total Straight Lines

4 + 4 + 3 = 11

Correct Answer: (B) 11

⚡ Quick Exam Shortcut

For minimum straight line questions:

1. Count continuous horizontal lines.
2. Count continuous lines parallel to the left side.
3. Count continuous lines parallel to the right side.
4. Add them together.

4 + 4 + 3 = 11

Shortcut Answer: 11 Straight Lines

🔍 Count the Triangles

The figure contains triangles of different sizes. Counting all the smallest, medium and larger composite triangles gives:

The triangles are:

AGE, EGC, GFC, BGF, DGB, ADG,
AGC, BGC, ABG, AFC, BEG, BDC,
ABF, ABE, DAC, ABC

Total number of triangles: 16

Correct Answer: (A) 16

⚡ Quick Exam Shortcut

In triangle-counting questions, do not stop after counting the visible small triangles. Always check for:

• Medium triangles formed by combining two small triangles.
• Large triangles formed by combining multiple regions.
• The complete outer triangle.

Total Triangles = 16

🔍 Analyze the Figure

This is a complex triangle-counting figure containing several overlapping triangles of different sizes.

To obtain the correct answer, count the triangles systematically according to their sizes.

Step 1: Count Smallest Triangles

First count all the elementary triangles formed inside the three squares and the outer triangular sections.

Small triangles = 24

Step 2: Count Medium-Sized Triangles

Next count the triangles formed by combining two or more small triangles within the individual sections of the figure.

Medium triangles = 10

Step 3: Count Largest Composite Triangles

Finally count the larger triangles formed by combining multiple sections of the figure, including those extending across the squares and the outer boundary.

Large triangles = 3

Step 4: Calculate Total Triangles

24 + 10 + 3 = 37

Correct Answer: (c) 37

⚡ Quick Exam Shortcut

In counting-of-figures questions, the most common mistake is counting only the visible small triangles and forgetting the larger composite triangles formed by combining adjacent regions.

After counting all small, medium and large triangles, the total number of triangles is: 37

Shortcut Answer: Total Triangles = 37

🔍 Analyze the Figure

Count the triangles according to their sizes.

Step 1: Small Triangles

8 Triangles

Step 2: Medium Triangles

6 Triangles

Step 3: Large Triangles

4 Triangles

Step 4: Largest Composite Triangles

2 Triangles

8 + 6 + 4 + 2 = 20

Correct Answer: (c) 20

⚡ Quick Exam Shortcut

Count triangles size-wise:

• Small = 8
• Medium = 6
• Large = 4
• Largest = 2

Total = 20

Shortcut Answer: 8 + 6 + 4 + 2 = 20

🔍 Analyze the Figure

The figure consists of a central square having both diagonals and centre lines, with four triangles attached on its four sides.

Step 1: Count Small Triangles

16 Triangles

Step 2: Count Medium Triangles

12 Triangles

Step 3: Count Large Triangles

8 Triangles

Step 4: Calculate Total

16 + 12 + 8 = 36

Correct Answer: (b) 36

⚡ Quick Exam Shortcut

Count triangles size-wise:

• Small = 16
• Medium = 12
• Large = 8

Total = 36

Shortcut Answer: 16 + 12 + 8 = 36

🔍 Analyze the Figure

Count triangles according to their sizes.

Step 1: Small Triangles

4 Triangles

Step 2: Medium Triangles

4 Triangles

Step 3: Large Triangles

2 Triangles

4 + 4 + 2 = 10

ABG, BCG, CGE, CDE, AGE, AEF, ABE, ABC, BCE, ACE

Correct Answer: (a) 10

⚡ Quick Exam Shortcut

Ignore the circle and count only the regions formed by the straight line segments.

• Small triangles = 4
• Medium triangles = 4
• Large triangles = 2

Total = 10

🔍 Analyze the Figure

Count the squares size-wise.

Step 1: Outer Figure

• 4 corner squares
• 1 outer square

Total = 5

Step 2: Middle Square Region

• 4 small squares inside the middle square
• 1 middle square

Total = 5

Step 3: Innermost Square Region

• 4 tiny squares
• 1 innermost square

Total = 5

Step 4: Adjust Overcount

One square is counted twice due to overlapping boundaries.

16 − 1 = 15

ABI, BIC, ALJ, CIJ, AHJ, CDJ, JHG, JDE, GJF, EJF, ABC, BCJ, ACJ, BAJ, AJG, CJE, GJE, ACG, ACE, CGE, AGE

Correct Answer: (c) 15

⚡ Quick Exam Shortcut

Count square-by-square:

• Outer figure = 5
• Middle square region = 5
• Inner square region = 5

Total: 5 + 5 + 5 = 15

Shortcut Answer: 15 Squares

🔍 Count the Rectangles

Step 1: Small Rectangles

• Top row = 3 rectangles
• Bottom row = 2 rectangles

Total = 5

Step 2: Top Row Combinations

• Left + Middle
• Middle + Right
• Left + Middle + Right

Total = 3

Running Total = 8

Step 3: Full-Height Rectangles

2 Rectangles

Running Total = 10

Step 4: Outer Rectangle

1 Rectangle

10 + 1 = 11

The rectangles are : ABCD, AEJK, AFIK, ABGK, EFIJ, EBGJ, FBGI, FBCH, IGCH, AFHD, KIHD

Correct Answer: (d) 11

⚡ Quick Exam Shortcut

Count rectangles row-wise and then add all larger combinations.

5 + 3 + 2 + 1 = 11

Shortcut Answer: 11 Rectangles

🔍 Count the Rectangles

Only the rectangular portion contributes to the rectangle count. The arrow-head section does not form any rectangle.

Step 1: Small Rectangles

4 (top row) + 4 (bottom row) = 8

Step 2: Two-Column Rectangles

3 (top row) + 3 (bottom row) = 6

Running Total = 14

Step 3: Three-Column Rectangles

2 (top row) + 2 (bottom row) = 4

Running Total = 18

The rectangles are : ABKJ; JKHI; BCLK; KLGH; CDML; LMFG; ACGI; ACLJ; JLGI; BDFH; BDMK; KMFH; ADFI; ADMJ; JMFI ABHI, BCGH and CDFG are squares.

We know that every square is a rectangle. But its reverse is not always true.

Correct Answer: (d) 18

⚡ Quick Exam Shortcut

Ignore the arrow-head and count only complete rectangles.

• Small rectangles = 8
• Two-column rectangles = 6
• Three-column rectangles = 4

8 + 6 + 4 = 18

Shortcut Answer: 18 Rectangles

🔍 Count the Circles

Count all complete circles visible in the figure.

Step 1: Outer Ring Circles

The outer ring contains 12 circles.

Total = 12

Step 2: Inner Ring Circles

The inner ring contains 6 circles.

12 + 6 = 18

Step 3: Centre Circle

There is 1 circle at the centre.

18 + 1 = 19

Step 4: Boundary Circles

The figure also contains:

• 1 outer enclosing circle
• 1 inner enclosing circle

19 + 2 = 21

Correct Answer: (d) 21

⚡ Quick Exam Shortcut

Count circles ring-wise:

• Outer Ring = 12
• Inner Ring = 6
• Centre = 1
• Boundary Circles = 2

12 + 6 + 1 + 2 = 21

Shortcut Answer: 21 Circles

🔍 Analysis

The diagonals are : EC, AC, BE, BF, AD, CF

Correct Answer: (d) 6

🔍 Analyze the Figure

The figure is formed by 7 regular hexagons arranged in a honeycomb pattern.

Step 1: Count the Outer Boundary Sides

The complete outer boundary consists of 24 sides.

Outer sides = 24

Step 2: Multiply by Side Length

Each side of a hexagon measures 5 cm.

Perimeter = 24 × 5

= 120 cm

Correct Answer: (C) 120 cm

⚡ Quick Exam Shortcut

For 7 hexagons arranged in a honeycomb pattern:

Outer Boundary = 24 Side Units

Perimeter = 24 × 5 = 120 cm

Shortcut Answer: 120 cm

1. Identify the figure to be counted.
2. Count the smallest figures first.
3. Count larger combined figures.
4. Check for hidden figures.
5. Use formulas wherever applicable.
6. Verify the final count.

• Missing larger combined figures.
• Double counting the same figure.
• Ignoring hidden shapes.
• Using random counting order.
• Skipping verification.

• Always count from smallest to largest.
• Use symmetry whenever possible.
• Apply shortcut formulas for squares and rectangles.
• Mark counted figures mentally.
• Verify your answer before final submission.

Identify Figure Type

↓

Count Small Figures

↓

Count Combined Figures

↓

Find Hidden Figures

↓

Verify Total Count