Counting of Figures

Counting of Figures is a topic where a complex figure is given and you have to count how many triangles, squares, rectangles or other shapes are present in it - including the hidden and overlapping ones. It is a topic that requires systematic counting to avoid missing any figure.

Basic Concept:

When counting figures, you must count:

• Small individual figures
• Medium figures made by combining 2 small figures
• Large figures made by combining 3 or more small figures
• The largest outer figure itself

Most students only count the small individual figures and miss the larger combined ones - that is the most common mistake.

Types of Counting of Figures Questions in SSC CGL:

Type 1 - Counting Triangles This is the most common type. A figure made of triangles is given and you have to count all triangles.

Example: 

A large triangle divided into 4 smaller triangles.

Count:

• Small triangles = 4 (individual ones)
• Medium triangles = 0 (no combination of 2 makes a new triangle here)
• Large triangle = 1 (the whole figure)
• Total = 5 triangles

Formula for triangles in a row: If a triangle is divided into n rows: Total triangles = n(n+2)(2n+1)/8

Type 2 - Counting Rectangles and Squares A grid figure is given and you have to count all rectangles and squares including overlapping ones.

Example: 

A 2×2 grid (2 rows and 2 columns)

Count:

• 1×1 squares = 4
• 1×2 rectangles (horizontal) = 2
• 2×1 rectangles (vertical) = 2
• 2×2 square = 1
• Total = 9

Formula for rectangles in m×n grid: Total rectangles = m(m+1)/2 × n(n+1)/2

For 2×2: 2×3/2 × 2×3/2 = 3×3 = 9 

Type 3 - Counting Squares Only Only squares (not rectangles) need to be counted.

Formula for n×n grid: Total squares = n² + (n-1)² + (n-2)² + ... + 1² = n(n+1)(2n+1)/6

For 3×3 grid: 9 + 4 + 1 = 14 squares

Systematic Method to Count Triangles:

Step 1 - Label each smallest triangle with a number (1, 2, 3...). 

Step 2 - Count all individual small triangles first. 

Step 3 - Count triangles made by combining 4 small triangles (medium size). 

Step 4 - Count the whole outer triangle. 

Step 5 - Add all counts together.

Example - Triangle with 3 rows (9 small triangles):

• Small triangles = 9 (T1 to T9)
• Medium triangles (4 units each) = 3
• Whole triangle = 1
• Total = 13

Quick Formulas - Must Memorize:

Triangles:

• Triangle divided into n rows = n(n+2)(2n+1)/8

Rectangles in m×n grid:

• Total = m(m+1)/2 × n(n+1)/2

Squares in n×n grid:

• Total = n(n+1)(2n+1)/6

Important Tips:

• Always label the smallest units before starting to count
• Never count the same figure twice
• Use a systematic approach - go from smallest to largest
• 2 to 3 questions come from this topic in SSC CGL every year
• Practice at least 10 counting questions daily