Figure Counting
Key Concepts
| # | Concept | Explanation |
|---|---|---|
| 1 | Triangle Counting | Count all possible triangles formed by intersecting lines. Use formula n(n+1)(n+2)/6 for triangles in a row. |
| 2 | Square Counting | Count squares of all sizes. For n×n grid: 1² + 2² + … + n² = n(n+1)(2n+1)/6 |
| 3 | Rectangle Counting | Count rectangles by selecting 2 horizontal and 2 vertical lines. Formula: C(m,2) × C(n,2) for m×n grid |
| 4 | Embedded Figures | Count smaller figures hidden within larger complex figures. Look for overlapping and shared boundaries |
| 5 | Pattern Recognition | Identify repeating patterns and count elements systematically row-wise or column-wise |
| 6 | Mirror Images | Count figures that are mirror images of each other. Check for symmetry axis |
| 7 | Rotation Counting | Count figures that are identical after rotation (90°, 180°, 270°) |
| 8 | Overlapping Figures | Count distinct figures when multiple shapes overlap. Use different colors mentally to separate |
15 Practice MCQs
- Count the number of triangles in the given figure: Question: How many triangles are there in the figure with 4 horizontal lines intersecting 4 vertical lines forming a grid?
- A) 16
- B) 20
- C) 24
- D) 28
Answer: B) 20 Solution: Using triangle counting formula for 4×4 grid: 4×5×6/6 = 20 triangles Shortcut: For n×n grid, use n(n+1)(n+2)/6 Concept: Triangle Counting
- Count squares in the figure: Question: How many squares are there in a 5×5 chessboard?
- A) 55
- B) 65
- C) 75
- D) 85
Answer: A) 55 Solution: 1² + 2² + 3² + 4² + 5² = 1 + 4 + 9 + 16 + 25 = 55 Shortcut: Sum of squares formula: n(n+1)(2n+1)/6 Concept: Square Counting
- Count rectangles in the grid: Question: How many rectangles can be formed in a 3×4 grid?
- A) 60
- B) 70
- C) 80
- D) 90
Answer: A) 60 Solution: C(4,2) × C(5,2) = 6 × 10 = 60 Shortcut: C(m+1,2) × C(n+1,2) for m×n grid Concept: Rectangle Counting
- Count triangles in complex figure: Question: Count triangles in a figure with a large triangle divided into 4 smaller triangles by lines from vertices to midpoint of opposite sides
- A) 5
- B) 8
- C) 10
- D) 12
Answer: C) 10 Solution: 4 small triangles + 3 medium triangles + 2 large triangles + 1 largest triangle = 10 Shortcut: Count systematically by size Concept: Triangle Counting
- Count embedded circles: Question: How many circles are completely hidden inside squares in the given figure?
- A) 3
- B) 4
- C) 5
- D) 6
Answer: B) 4 Solution: Look for circles whose entire boundary is within square boundaries Shortcut: Trace each circle’s boundary mentally Concept: Embedded Figures
- Count overlapping squares: Question: Two squares overlap such that their centers coincide and one is rotated 45°. How many distinct regions are formed?
- A) 6
- B) 8
- C) 10
- D) 12
Answer: B) 8 Solution: The overlapping creates 8 distinct triangular regions Shortcut: Draw and shade different regions Concept: Overlapping Figures
- Count pattern repetition: Question: In a sequence of 50 figures where the pattern △○□ repeats, how many triangles are there?
- A) 15
- B) 16
- C) 17
- D) 18
Answer: C) 17 Solution: 50 ÷ 3 = 16 complete cycles + 1 extra figure (triangle) Shortcut: Divide total by pattern length Concept: Pattern Recognition
- Count mirror images: Question: How many pairs of mirror images exist in a row of 8 identical but differently oriented arrows?
- A) 2
- B) 3
- C) 4
- D) 5
Answer: C) 4 Solution: Each arrow can have a mirror image except the middle one in odd numbers Shortcut: n/2 for even n, (n-1)/2 for odd n Concept: Mirror Images
- Count rotationally symmetric figures: Question: In a circle divided into 8 equal sectors with alternating patterns, how many figures are identical after 90° rotation?
- A) 2
- B) 3
- C) 4
- D) 6
Answer: C) 4 Solution: 360° ÷ 90° = 4, so 4 figures will match after rotation Shortcut: Divide 360 by rotation angle Concept: Rotation Counting
- Count triangles in star figure: Question: A 5-pointed star (pentagram) has how many triangles?
- A) 5
- B) 10
- C) 15
- D) 20
Answer: B) 10 Solution: 5 small triangles + 5 larger triangles = 10 Shortcut: Count points and intersections Concept: Complex Figure Counting
- Count squares in nested squares: Question: A square is divided into 4 smaller squares, and this process is repeated once more. Total squares?
- A) 20
- B) 21
- C) 25
- D) 30
Answer: B) 21 Solution: 1 (large) + 4 (medium) + 16 (small) = 21 Shortcut: Sum of geometric progression Concept: Nested Figures
- Count hexagons in honeycomb: Question: A honeycomb pattern with 3 rows and 4 columns of hexagons has how many hexagons?
- A) 10
- B) 12
- C) 14
- D) 16
Answer: B) 12 Solution: 3 × 4 = 12 hexagons Shortcut: Simple multiplication for regular patterns Concept: Pattern Counting
- Count parallelograms: Question: In a figure with 3 parallel horizontal lines and 4 parallel vertical lines, how many parallelograms?
- A) 18
- B) 24
- C) 30
- D) 36
Answer: A) 18 Solution: C(3,2) × C(4,2) = 3 × 6 = 18 Shortcut: Same as rectangle counting Concept: Parallelogram Counting
- Count figures with common area: Question: Three circles intersect pairwise. How many common areas are shared by at least two circles?
- A) 3
- B) 4
- C) 6
- D) 7
Answer: B) 4 Solution: 3 pairwise intersections + 1 common to all three Shortcut: Draw Venn diagram mentally Concept: Overlapping Figures
- Count triangles in complex grid: Question: A triangle is subdivided by drawing lines from each vertex to points that trisect the opposite sides. Total triangles?
- A) 13
- B) 15
- C) 17
- D) 19
Answer: C) 17 Solution: Count systematically: 9 smallest + 6 medium + 2 large = 17 Shortcut: Count by size categories Concept: Complex Triangle Counting
Speed Tricks
| Situation | Shortcut | Example |
|---|---|---|
| Triangle in Row | n(n+1)(n+2)/6 | 5 rows: 5×6×7/6 = 35 triangles |
| Square in Grid | Sum of squares | 4×4 grid: 1²+2²+3²+4² = 30 |
| Rectangle Counting | C(m+1,2)×C(n+1,2) | 3×4 grid: C(4,2)×C(5,2) = 6×10 = 60 |
| Overlapping Circles | n(n-1)/2 + 1 | 3 circles: 3×2/2 + 1 = 4 regions |
| Pattern Repetition | Total ÷ Pattern length | 100 figures, pattern length 5: 100÷5 = 20 cycles |
Quick Revision
| Point | Detail |
|---|---|
| 1 | Always count systematically - smallest to largest or vice versa |
| 2 | For triangles: count by size (small, medium, large) |
| 3 | For squares: remember sum of squares formula n(n+1)(2n+1)/6 |
| 4 | For rectangles: use combination formula C(m,2)×C(n,2) |
| 5 | Mark counted figures mentally to avoid double counting |
| 6 | Look for symmetry - reduces counting effort by half |
| 7 | In overlapping figures, count distinct regions separately |
| 8 | For complex figures, break into simpler components |
| 9 | Practice visualization - mentally shade different regions |
| 10 | Time limit: Spend max 45 seconds per figure counting question |
BETA
