Counting of Figures

 Counting of Figures is a visual reasoning problem where we count specific geometric shapes (triangles, squares, rectangles, etc.) embedded within a complex figure. It tests observation skills, pattern recognition, and systematic counting.

🔍 Types of Counting Questions

Type	Example
✅ Count Triangles	Single, overlapping, divided triangles
✅ Count Squares	Small, large, overlapping squares
✅ Count Rectangles	Formed by horizontal and vertical lines
✅ Count Circles/Semicircles	Often asked rarely in exams
✅ Count Lines	Straight and intersecting lines

🛠️ Important Methods and Tricks

✅ A. Labelling Trick

Label corners or segments (A, B, C...) to track shapes visually.

✅ B. Use Shape Formulas (Shortcuts)

🔺 Triangles in a Series (Like a Triangle Made of Lines)

Formula:

$$\text{Total triangles} = n(n+1)/2$$

Where n = number of divisions in base

🔳 Squares in a Grid (Like 2x2 or 3x3 box)

Formula:

$$\text{Total squares} = n^2 + (n-1)^2 + (n-2)^2 + \dots + 1$$

🧩 Solved Examples with Diagrams

📐 Example 1: Count Triangles

Diagram:

     A
    / \
   /___\
  B     C

🧠 Count:

• △ABC (big triangle)

• △AB, △BC (2 small)
  ✅ Total = 3 triangles

📐 Example 2: Count Squares in Grid

Diagram: 2x2 Square Grid

┌───┬───┐
│ 1 │ 2 │
├───┼───┤
│ 3 │ 4 │
└───┴───┘

🧠 Count:

• 4 small squares

• 1 large square (2x2)

✅ Total = 5 squares

📐 Example 3: Count Rectangles in Grid

Diagram: 2 Rows × 3 Columns

┌───┬───┬───┐
│   │   │   │
├───┼───┼───┤
│   │   │   │
└───┴───┴───┘

🧠 Use Formula:
Total Rectangles = (rows × (rows + 1)/2) × (columns × (columns + 1)/2)

Here: 2 rows, 3 columns

$$= (2×3/2) × (3×4/2) = 3 × 6 = 18$$

✅ Total = 18 rectangles

📐 Example 4: Count All Triangles in Star

Diagram (Star made by overlapping two triangles):

   /\
  /__\
  \  /
   \/

🧠 Count:

• 2 large triangles

• 6 small overlapping triangles

• 1 central hexagon (not a triangle)

✅ Total = 8 triangles

📐 Example 5: Complex Figure

Diagram:

   A
  /|\
 /_|_\
B__|__C

🧠 Count:

• Small triangles = 3

• Large triangles (combinations) = 2
  ✅ Total = 5 triangles

💡Tips and Shortcuts

Tip Type	Description
👁️ Visual Scanning	Start from small to large shapes
🔄 Elimination	Cross out counted shapes to avoid duplication
➕ Addition Trick	Count individual + combo shapes
🧠 Memorize Formulas	Especially for grid and series questions
🖊️ Label Everything	Mark corners or boxes A, B, C... for accuracy

📘 Practice Questions – Counting of Figures

Question 1: Counting Triangles

Diagram:

    ▲
   / \
  /   \
 /_____\
   \ /
    ▼

Answer:

• Small triangles = 4 (top, left, right, bottom)

• Large triangle = 1

• Total = 5 triangles

Question 2: Counting Squares

Diagram:

┌───┬───┐
│   │   │
├───┼───┤
│   │   │
└───┴───┘

Answer:

• 1x1 squares = 4

• 2x2 squares = 1

• Total = 5 squares

Question 3: Counting Rectangles

Diagram:

┌───┬───┐
│   │   │
└───┴───┘

Answer:

• Single rectangles = 4 (2 vertical, 2 horizontal)

• Combined rectangles = 2 (top row, bottom row)

• Whole figure = 1

• Total = 7 rectangles

Question 4: Counting Triangles in a Star

Diagram:

    ▲
   / \
  /   \
◀-----▶
  \   /
   \ /
    ▼

Answer:

• Small triangles = 8 (4 upward, 4 downward)

• Large triangles = 2 (top and bottom)

• Total = 10 triangles

Question 5: Counting Cubes in a 3D Figure

Diagram: (Stack of 4 cubes: 2 base, 2 top)

  ◼◼
  ◼◼

Answer:

• Single cubes = 4

• Pairs = 4 (2 horizontal, 2 vertical)

• All 4 cubes = 1

• Total = 9 cubes

Question 6: Counting Parallelograms

Diagram:

  /\/\
 /\/\/\

Answer:

• Single parallelograms = 6

• Double parallelograms = 2

• Total = 8 parallelograms

Question 7: Counting Circles in a Venn Diagram

Diagram: (3 overlapping circles: A, B, C)

text

Copy

Download

  ⚪⚪⚪
 ⚪   ⚪
  ⚪⚪⚪

Answer:

• Single circles = 3

• Double overlaps = 3 (A+B, B+C, A+C)

• Triple overlap = 1 (A+B+C)

• Total = 7 circle segments

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Question 8: Counting Hexagons in a Honeycomb

Diagram: (7 hexagons: 1 centre + 6 surrounding)

  ⬢⬢⬢
 ⬢⬢⬢
  ⬢⬢

Answer:

• Small hexagons = 7

• Large hexagon (whole figure) = 1

• Total = 8 hexagons

Question 9: Counting Rectangles in a 3x3 Grid

Diagram:

┌─┬─┬─┐
│ │ │ │
├─┼─┼─┤
│ │ │ │
├─┼─┼─┤
│ │ │ │
└─┴─┴─┘

Answer:

• Formula: C(4,2) × C(4,2) = 6 × 6 = 36 rectangles

Question 10: Counting Triangles in a Complex Figure

Diagram:

    ▲
   /|\
  / | \
 /__|__\
    |
    |

Answer:

• Left small triangles = 2

• Right small triangles = 2

• Central triangles = 2 (upper and lower)

• Large triangle = 1

• Total = 7 triangles