Counting of Figures Notes for SSC Exams
Rahul Kumar
SSC Exam Expert & Content Editor
In this note
- 1.Formulas to Count Figures
- 2.Squares and Rectangles in a Grid
- 3.Triangles in a Triangle
- 4.Triangles When Each Side Is Divided Equally
- 5.Solved Examples
- 6.Tips for Counting
- 7.Quick Tips
- 8.Quick Revision Points
- 9.Practice Questions With Answers
- 10.How to Practise This Chapter
- 11.Frequently Asked Questions
PDF Notes: Counting of Figures Notes for SSC Exams
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Counting of figures questions ask you to count the triangles, squares or rectangles in a given figure. Counting one by one wastes time and gives mistakes, but a few formulas make most questions easy. These counting of figures notes for SSC give the formulas, a table of answers and solved examples.
Formulas to Count Figures
| Figure | Formula |
|---|---|
| Squares in an n × n grid | n(n + 1)(2n + 1) / 6 |
| Rectangles (including squares) in an m × n grid | [m(m + 1) / 2] × [n(n + 1) / 2] |
| Rectangles in an n × n grid | [n(n + 1) / 2] squared |
| Triangles in a triangle with n lines drawn from one vertex to the opposite side | (n + 1)(n + 2) / 2 |
| Triangles when each side of a triangle is divided into n equal parts and lines are drawn parallel to the sides | n(n + 2)(2n + 1) / 8 (take the whole number part) |
Squares and Rectangles in a Grid
| Grid | Squares | Rectangles (including squares) |
|---|---|---|
| 1 × 1 | 1 | 1 |
| 2 × 2 | 5 | 9 |
| 3 × 3 | 14 | 36 |
| 4 × 4 | 30 | 100 |
| 5 × 5 | 55 | 225 |
| 8 × 8 (chessboard) | 204 | 1296 |
Rectangles that are not squares = total rectangles − squares. On a chessboard this is 1296 − 204 = 1092.
Triangles in a Triangle
| Lines from one vertex | Total triangles |
|---|---|
| 1 | 3 |
| 2 | 6 |
| 3 | 10 |
| 4 | 15 |
| 5 | 21 |
Triangles When Each Side Is Divided Equally
| Parts on each side (n) | Total triangles |
|---|---|
| 1 | 1 |
| 2 | 5 |
| 3 | 13 |
| 4 | 27 |
Solved Examples
Example 1. How many squares are there in a 3 × 3 grid?
1² + 2² + 3² = 1 + 4 + 9 = 14.
Example 2. How many rectangles (including squares) are there in a grid of 3 rows and 4 columns?
[3 × 4 / 2] × [4 × 5 / 2] = 6 × 10 = 60.
Example 3. A triangle has 3 lines drawn from one vertex to the opposite side. How many triangles are there?
(3 + 1)(3 + 2) / 2 = 4 × 5 / 2 = 10.
Example 4. How many squares are there in a 5 × 5 grid?
1 + 4 + 9 + 16 + 25 = 55.
Tips for Counting
- Count the smallest figures first, then the figures made of two, three and more parts.
- Label the parts with letters and write the combinations.
- Count a figure only once. Do not count the same triangle again.
- In a grid, use the formula instead of counting.
Quick Tips
- Learn the squares and rectangles table for grids up to 5 × 5.
- Count from the smallest to the largest figure.
- Label the parts with letters to avoid counting twice.
- Use the formula whenever the figure is a regular grid.
Quick Revision Points
- Squares in an n × n grid = 1² + 2² + ... + n².
- Rectangles in an m × n grid = [m(m+1)/2] × [n(n+1)/2].
- A chessboard has 204 squares and 1296 rectangles.
- Triangles with n lines from one vertex = (n+1)(n+2)/2.
- Count each figure only once.
Practice Questions With Answers
- How many squares are there in a 2 × 2 grid?
Answer: 5 - How many squares are there in a 4 × 4 grid?
Answer: 30 - How many rectangles (including squares) are there in a 2 × 2 grid?
Answer: 9 - How many rectangles (including squares) are there in a grid of 3 rows and 2 columns?
Answer: 18 - How many triangles are there if 4 lines are drawn from one vertex of a triangle to the opposite side?
Answer: 15 - How many squares are there on a chessboard?
Answer: 204 - How many rectangles, including squares, are there on a chessboard?
Answer: 1296 - Each side of a triangle is divided into 3 equal parts and lines are drawn parallel to the sides. How many triangles are there?
Answer: 13 - How many squares are there in a 5 × 5 grid?
Answer: 55 - What is the formula for the number of squares in an n × n grid?
Answer: n(n + 1)(2n + 1) / 6
How to Practise This Chapter
Practise figure questions daily on paper, because you must see the figures to learn them. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Non Verbal Reasoning notes page, and also read the Reasoning notes and Verbal Reasoning notes. Read the previous topic, Mirror and Water Image, and the next topic, Paper Folding and Cutting.
Frequently Asked Questions
What are counting of figures questions?
They are questions in which you count the triangles, squares or rectangles in a given figure.
How many squares are there in an n by n grid?
The number is n(n + 1)(2n + 1) / 6.
How many rectangles are there in an m by n grid?
The number is [m(m + 1) / 2] times [n(n + 1) / 2], including the squares.
How many squares are there on a chessboard?
There are 204 squares on a chessboard.
How many rectangles are there on a chessboard?
There are 1296 rectangles, including the 204 squares.
How many triangles are there with n lines from one vertex?
The number is (n + 1)(n + 2) / 2.
How do I avoid counting a figure twice?
Label the parts with letters and write each combination only once.
Where should I start counting?
Start with the smallest figures and then count the larger ones.
Is this topic asked in SSC exams?
Yes. Counting of figures is asked in SSC reasoning papers.
How much time should I take for one question?
About 40 to 60 seconds, as this is a slower topic.
What are common mistakes?
Missing the large figures made of many parts and counting the same figure twice.
How many questions come from this topic in SSC exams?
Usually one question, but the number changes from year to year.
Previous chapter
Mirror and Water Image Notes for SSC Exams
Next chapter
Paper Folding and Cutting Notes for SSC Exams
Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.