Counting Figures
🔒 Log in to trackCount triangles, squares, rectangles or straight lines in a figure. Recognise the standard shapes first (square with diagonals = 8, with midlines too = 16; apex lines (k+1)(k+2)/2; grid rectangles C(m+1,2)C(n+1,2); squares 1² + … + n²), and count anything irregular size by size so nothing is missed or counted twice.
One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.
Every formula on one printable page, grouped by subtopic.
4 exam-level questions worked step by step.
57 questions — untimed practice or a timed test with analysis.
Track record in the exam
Questions per shift in recent SSC CGL papers.
Test difficulty mix (57 questions)
Question patterns exams keep repeating
Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.
Triangle with apex lines
very commonA triangle with several lines from the top corner to the base, sometimes with lines parallel to the base.
How to solve: Count the apex lines k and the parallel cuts h. Use (k+1)(k+2)/2 × (h+1). No need to count by eye.
Example: A triangle has 4 lines from the apex to the base. How many triangles?
(5 × 6)/2 = 15.
Square/rectangle with diagonals
very commonA square or rectangle with diagonals and midlines, or a row of such squares.
How to solve: Count by size: smallest pieces, then 2-piece triangles, then half-squares. Square with diagonals = 8; with midlines too = 16.
Example: How many triangles are in a square with both diagonals?
4 small + 4 half-squares = 8.
Triangle divided into rows
commonA big triangle made of small upright and upside-down triangles.
How to solve: Use 1, 5, 13, 27, 48 for 1 to 5 rows. Do not forget the upside-down triangles.
Example: A triangle is split into 4 rows of small triangles. How many triangles?
Standard value for 4 rows = 27.
Irregular composite figure
commonHouse shapes, pentagon with a star, two overlapping triangles, a triangle with medians.
How to solve: Label corners and crossings; count triangles part by part, then the ones crossing parts. Known values: medians 16, pentagon with diagonals 35.
Example: How many triangles are formed when all three medians of a triangle are drawn?
6 small + 3 two-piece + 6 half-triangles + 1 whole = 16.
Squares in a grid
commonA chessboard-style grid; count squares.
How to solve: Add (m−k+1)(n−k+1) for every size k. For n × n: 1² + … + n².
Example: How many squares are there on a 5 × 5 grid?
1 + 4 + 9 + 16 + 25 = 55.
Rectangles in a grid
occasionalA grid; count all rectangles, squares included.
How to solve: Choose 2 vertical and 2 horizontal lines: C(m+1,2) × C(n+1,2).
Example: How many rectangles are there in a 3 × 3 grid?
C(4,2) × C(4,2) = 6 × 6 = 36.
Tilted and irregular squares
occasionalA diamond drawn inside a square or grid, or cells arranged in a plus or L shape.
How to solve: Count upright squares first, then add each tilted square whose four sides are drawn. For irregular grids, check all four sides of each candidate.
Example: A 2 × 2 grid has a diamond joining the middles of its outer sides. How many squares?
4 + 1 upright + 1 tilted = 6.
Straight lines count
occasionalHow many straight lines are there, or the least number of lines needed to draw a figure.
How to solve: Sweep direction by direction; pieces on one straight path count once. Pentagram = 5, grid = (m+1)+(n+1).
Example: How many straight lines are there in a square with both diagonals?
Horizontal 2 + vertical 2 + slanting 2 = 6.