Core ConceptRead this first — the foundation of the topic
CORE CONCEPT
You are given a complex figure made of triangles, squares, rectangles, lines, or other shapes. Your job is to count ALL shapes of a specific type (or all shapes total) without missing any or double-counting
KEY RULES
Count systematically — start from one corner and move in order (left to right, top to bottom).
2. Count individual shapes first, then combinations of 2, 3, 4 shapes, etc.
3. Overlapping shapes count as separate shapes.
4. Don't recount the same shape twice when moving to combinations.
5.
Use different colors or marks mentally to track what you've counted.
Exam PatternsWhat examiners ask — read before attempting PYQs
— Count total triangles in a figure
— Count total squares/rectangles
— Count total line segments
— Count figures formed by combining smaller shapes
— Questions ask: "How many triangles are there?" or "Find the total number of squares."
SHORTCUT/TRICK:
Use the "Layer Method" — Count shapes by size. First count the smallest individual shapes. Then count shapes made of 2 units, then 3 units, and so on. This prevents overlap and ensures nothing is missed.
Worked ExampleSolve this step-by-step before moving on
1
Step 1
Count 1×1 squares = 4 (the four small squares)
2
Step 2
Count 2×2 squares = 1 (the entire large square)
3
Step 3
Add them up = 4 + 1 = 5 total squares
Answer: 5 squares
Exam TrapsCommon mistakes students make — avoid these
Students often count only the smallest individual shapes and forget to count larger shapes formed by combining smaller ones. For example, in a 2×2 grid, many count only the 4 small squares and give 4 as the answer, missing the 1 large square. Always count combinations systematically.
FORMULA (For grid-based counting):
For an n×n grid of squares:
Total = 1² + 2² + 3² + ... + n²
Example: For a 3×3 grid = 1 + 4 + 9 = 14 squares
Key Points to Remember
Count shapes systematically from one direction to avoid missing or double-counting.
Always count individual shapes first, then combinations of 2, 3, 4 units.
For an n×n grid of squares, use formula: Total = 1² + 2² + 3² + ... + n²
Overlapping shapes and shapes formed by combinations are counted separately.
Common error: forgetting to count larger shapes made from combining smaller ones.
Mark mentally or use colors to track which shapes you've already counted.
Exam-Specific Tips
In a 2×2 square grid, total number of squares = 5 (four 1×1 squares + one 2×2 square).
In a 3×3 square grid, total number of squares = 14 (calculated as 1² + 2² + 3² = 1 + 4 + 9).
For a figure with straight lines, each intersection point can create multiple segments to count.
In triangle counting problems, overlapping triangles of different sizes must be counted as separate entities.
The sum formula for square counting in n×n grids is: n(n+1)(2n+1)/6 for advanced calculations.
SSC CGL typically presents 2-3 figure counting questions in the Non-Verbal Reasoning section.
Time limit per counting figure question is approximately 1-1.5 minutes in actual exam.
Grid-based counting (squares and rectangles) appears more frequently than line segment counting.
Practice MCQs
Counting Figures — Practice Questions
12graded MCQs · easy to hard · full solution & trap analysis
A square is divided into 16 equal smaller squares (4×4 grid). How many squares of all sizes can be counted in this figure?
Practice 2medium
A figure consists of a hexagon with all its diagonals drawn. How many triangles are formed in total?
Practice 3hard
In a complex figure, there are 5 concentric circles with 8 radial lines dividing each circle into 8 equal sectors. Some sectors are shaded, some are unshaded. The shading pattern follows: Circle 1 (innermost) has 3 consecutive shaded sectors. Circle 2 has 5 non-consecutive shaded sectors. Circle 3 has 4 shaded sectors in alternating pattern. Circle 4 has 2 shaded sectors opposite each other. Circle 5 (outermost) has all 8 sectors shaded. Count the total number of unshaded sectors across all circles.
Practice 4hard
A figure shows a large square subdivided by 3 horizontal and 3 vertical lines (creating a 4×4 grid of 16 small squares). Two diagonal lines are drawn from opposite corners. Additionally, a circle is drawn such that it passes through the centers of 4 corner squares (one circle, not 4 separate circles). How many distinct regions are formed by all these lines and the circle, where a region is defined as an area bounded by grid lines, diagonals, and/or the circle arc?
Practice 5hard
A figure shows a hexagon divided into 6 equilateral triangles (arranged in a honeycomb pattern around a central point). Each of the 6 triangles is further subdivided into 4 smaller equilateral triangles (by connecting midpoints of sides). Some of the smallest triangles are shaded. The shading pattern is: In triangles 1-3 (top half), all 4 smallest triangles are shaded. In triangles 4-6 (bottom half), only the central smallest triangle is shaded. Count the total number of unshaded smallest triangles.
Practice 6hard
In a complex figure, there is a large square divided into 4 equal smaller squares. Each of these 4 squares is further divided into 4 equal tiny squares. Additionally, there are 3 circles drawn such that each circle passes through the corners of exactly 2 adjacent smaller squares. How many total enclosed regions (squares + circular segments) are formed in this figure?
Practice 7hard
A figure consists of a 3×3 grid of squares. A diagonal line is drawn from the top-left corner to the bottom-right corner. Another diagonal line is drawn from the top-right corner to the bottom-left corner. These two diagonals intersect at the center. How many distinct triangular regions are formed by these diagonals and the grid lines?
Practice 8hard
A figure shows a rectangle with 5 horizontal lines and 4 vertical lines drawn inside it (including the borders). Additionally, 2 diagonal lines are drawn: one from the top-left to bottom-right, and one from top-right to bottom-left. These diagonals intersect all grid cells. How many quadrilateral regions (including squares and rectangles) are formed, excluding any regions that contain a diagonal line passing through them?
Practice 9hard
A figure contains a hexagon with all 6 sides equal. Inside, 3 non-intersecting diameters are drawn connecting opposite vertices. These diameters divide the hexagon into 6 triangular regions. Additionally, 3 circles are inscribed such that each circle is tangent to exactly 2 sides of the hexagon and passes through the center. How many distinct regions (including partial regions created by circle arcs) are formed in total?
Practice 10hard
A figure consists of overlapping triangles and squares arranged in a grid pattern. The grid is 4×4. Each cell contains either a triangle, a square, or both shapes overlapping. Count the total number of individual triangles (including those that form parts of composite figures) visible in the entire figure.
Practice 11hard
A figure shows a 3×3 grid of squares. Inside each square, there are smaller shapes: some squares contain 2 triangles, some contain 3 circles, some contain 1 square, and some contain a combination. The distribution is: Top row (L to R): 2 triangles, 3 circles, 2 triangles. Middle row: 3 circles, 1 square, 3 circles. Bottom row: 2 triangles, 1 square, 2 triangles. Additionally, there is 1 large triangle encompassing the entire 3×3 grid. Count the total number of triangles in the figure.
Practice 12hard
A figure consists of a 5×5 grid where each cell is either empty, contains a dot, or contains a line segment. The pattern is: Dots appear in cells (1,1), (1,5), (3,3), (5,1), (5,5). Line segments connect: (1,1)-(1,5), (1,5)-(5,5), (5,5)-(5,1), (5,1)-(1,1). Additionally, diagonals are drawn: (1,1)-(5,5) and (1,5)-(5,1). Count the total number of distinct triangles formed by the dots and line segments (including the diagonals).
60-Second Revision — Counting Figures
Remember: Count systematically by size — small shapes first, then combinations. Never skip larger combined shapes.
Formula for n×n square grid: Total = 1² + 2² + 3² + ... + n². For 3×3 grid = 14 squares.
Trap: Students miss composite shapes. In a divided square, the outer boundary also counts as one shape.
Layer method works best: identify all 1-unit shapes, then 2-unit shapes, then larger combinations.
Verify your count: Use a different mental path to recount. If answers match, you're likely correct.
Mark shapes mentally as you count to avoid recounting or skipping in complex figures.
Practice on grid patterns (2×2, 3×3, 3×4) — these dominate SSC CGL counting questions.