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.
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Exam Patterns
What 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.
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Worked Example
Solve 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
COMMON MISTAKE:
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
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.