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 lines from the top
very commonA triangle with lines drawn from the top corner to the base.
Count the lines k drawn inside from the top corner.
Triangles = (k + 1)(k + 2) ÷ 2. If h lines are parallel to the base, multiply by (h + 1).
A triangle has 4 lines drawn from its top corner to the base. How many triangles are there?
4 lines inside make 5 parts along the base
Triangles = (4 + 1)(4 + 2) ÷ 2
5 × 6 ÷ 2 = 15
Answer: 15
Square with diagonals
very commonA square or rectangle with diagonals or middle lines drawn inside.
Count the small triangles first, then bigger ones made of 2 parts.
Square with both diagonals = 8.
How many triangles are there in a square with both diagonals drawn?
4 small triangles (the 4 parts)
4 bigger triangles, each made of 2 parts (half the square)
4 + 4 = 8
Answer: 8 triangles
Big triangle made of rows
commonA big triangle made of many small triangles, some upright and some upside down.
Count upright triangles of each size, then the upside-down ones.
Totals for 1 to 5 rows: 1, 5, 13, 27, 48.
A big triangle is split into 4 rows of small triangles. How many triangles are there?
Upright: 10 small + 6 of size 2 + 3 of size 3 + 1 of size 4 = 20
Upside down: 6 small + 1 of size 2 = 7
20 + 7 = 27
Answer: 27
Mixed shapes
commonHouse shapes, a pentagon with a star, or a triangle with medians.
Name the corners and count triangles part by part.
Known totals: triangle with 3 medians = 16, pentagon with all diagonals = 35.
How many triangles are formed when all three medians of a triangle are drawn?
Smallest: 6
Made of 2 parts: 3. Made of 3 parts: 6
Whole triangle: 1. Total = 6 + 3 + 6 + 1
Answer: 16
Squares in a grid
commonA chessboard-style grid, and you must count all squares.
Count squares of each size: 1 × 1, 2 × 2, 3 × 3 and so on.
An n × n grid has 1² + 2² + ... + n² squares.
How many squares are there on a 5 × 5 grid?
1 × 1 squares: 25
2 × 2: 16, 3 × 3: 9, 4 × 4: 4, 5 × 5: 1
25 + 16 + 9 + 4 + 1
Answer: 55
Rectangles in a grid
occasionalA grid, and you must count all rectangles (squares count too).
A rectangle needs 2 vertical and 2 horizontal lines.
Rectangles = (ways to pick 2 vertical lines) × (ways to pick 2 horizontal lines).
How many rectangles are there in a 3 × 3 grid?
A 3 × 3 grid has 4 vertical and 4 horizontal lines
Ways to pick 2 of 4 lines = 6
6 × 6
Answer: 36
Tilted squares
occasionalA diamond drawn inside a square, or an irregular grid.
Count the upright squares first.
Then add each tilted square whose four sides are all drawn.
A 2 × 2 grid has a diamond joining the middles of its four outer sides. How many squares are there?
Upright squares: 4 small + 1 big = 5
Tilted square: the diamond = 1
5 + 1
Answer: 6
Straight lines
occasional'How many straight lines are there in the figure?'
Count direction by direction.
Parts that lie on one straight path count as one line.
How many straight lines are there in a square with both diagonals?
Horizontal lines: 2 (top and bottom)
Vertical lines: 2 (left and right)
Slanting lines: 2 (the diagonals)
Answer: 6