Counting Figures and Dot Situation
Counting Figures and Dot Situation
Counting questions ask how many triangles, squares or rectangles a figure contains. Count systematically, by size, and use the standard formulas for regular figures. Dot-situation questions ask you to find the option where dots can be placed in the same regions (inside some shapes, outside others) as in the given figure.
1. Formula Box
| Figure | Count |
|---|---|
| Triangle with k lines drawn from the top vertex to the base | (k + 1)(k + 2)/2 triangles |
| Square with both diagonals | 8 triangles |
| Square with both diagonals and both mid-lines | 16 triangles |
| Rectangle cut into n strips (one direction) | n(n + 1)/2 rectangles |
| m × n grid of unit squares | Rectangles = [m(m + 1)/2] × [n(n + 1)/2] |
| n × n grid | Squares = 1² + 2² + ... + n² |
| Stack: layers of 1, 4, 9, 16 cubes | Total = sum of the layers |
Count by size: smallest pieces first, then pieces made of 2, 3 and more
Dot situation: write the condition for each dot (inside which shapes, outside which), then test each option
✗ A triangle with 3 lines from the apex has 4 triangles | ✓ Count combined pieces too: (3 + 1)(3 + 2)/2 = 10
✗ A 4 × 4 board has 16 squares | ✓ Count larger squares too: 16 + 9 + 4 + 1 = 30
हिंदी नोट: आकृतियाँ गिनते समय आकार के क्रम में गिनिए: पहले सबसे छोटे, फिर दो-दो मिलाकर बने। बिंदु स्थिति में हर बिंदु की शर्त लिखिए: किन आकृतियों के अंदर, किनके बाहर।
Exam Pointer: These patterns had no verified NTPC shift in our check (triangle-counting and dot pages found were from DSSSB, SSC and police papers) and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Counting triangles
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A triangle has 3 lines drawn from its top vertex to its base, making 4 small triangles side by side. How many triangles are there in all?
(3 + 1)(3 + 2)/2 = 10: four single, three double, two triple and one whole.
Answer: 10
EXAMATLAS LEVEL
Q. A square has both diagonals and both lines joining the midpoints of opposite sides drawn. How many triangles does it contain?
There are 8 smallest triangles. Two small triangles from neighbouring quarters join into a triangle with a full side of the square as base and the centre as apex: 4. Each diagonal splits the square into 2 triangles: 4 in all. Total 8 + 4 + 4 = 16.
Answer: 16
Pattern 2: Counting squares and rectangles
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. How many squares are there on a 4 × 4 board?
1 × 1: 16, 2 × 2: 9, 3 × 3: 4, 4 × 4: 1, a total of 30.
Answer: 30
EXAMATLAS LEVEL
Q. A rectangle is divided into a grid of 3 columns and 2 rows of equal squares. How many rectangles (including squares) and how many squares does it contain?
Rectangles = (3 × 4/2) × (2 × 3/2) = 6 × 3 = 18. Squares: six 1 × 1 and two 2 × 2, a total of 8.
Answer: 18 rectangles; 8 squares
Pattern 3: Dot situation
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. In the given figure a dot lies inside the circle and the square but outside the triangle. In which option can a dot be placed in the same way? (a) A circle and a square overlap, and a small triangle lies inside the circle away from the square. (b) A circle lies inside a square and a triangle covers the whole circle. (c) A circle and a square lie apart, with a triangle between them. (d) A square lies inside a triangle and a circle overlaps the square.
(a) has a circle-and-square region outside the triangle. In (b) and (d), every point common to the circle and square is inside the triangle, and in (c) the circle and square do not meet.
Answer: (a)
EXAMATLAS LEVEL
Q. In the given figure one dot lies in the circle and the triangle only, and another lies in the square only. Which option allows both? (a) A circle, a square and a triangle overlapping like a three-set Venn diagram, each pair sharing a region outside the third. (b) A triangle inside a square, with a circle overlapping the triangle inside the square. (c) A circle and a triangle apart, with a square overlapping the circle. (d) A square inside a circle, with a triangle overlapping the circle outside the square.
(a) has both regions. In (b), the circle-triangle overlap lies inside the square, so it is not "circle and triangle only". In (c), the circle and triangle never meet. In (d), the square lies inside the circle, so there is no "square only" region.
Answer: (a)
Pattern 4: Counting cubes in a stack
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A stack has layers of 3 × 3, 2 × 2 and 1 × 1 cubes, one on top of another. How many cubes are there?
9 + 4 + 1 = 14.
Answer: 14
EXAMATLAS LEVEL
Q. A pyramid-shaped stack has four layers of 4 × 4, 3 × 3, 2 × 2 and 1 × 1 cubes, centred one on another. How many cubes does it have, and how many can be seen from directly above?
16 + 9 + 4 + 1 = 30 cubes. From above, every position of the 4 × 4 base shows exactly one cube (the topmost there), so 16 are seen.
Answer: 30; 16
60-Second Revision
- Count by size, smallest first.
- Triangle with k apex lines: (k + 1)(k + 2)/2.
- Grid rectangles: [m(m + 1)/2] × [n(n + 1)/2]; n × n squares: sum of squares.
- Dot situation: write each dot's in-out condition, then test options.