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
1. 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) 28Answer: 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
2. Count squares in the figure:
**Question:** How many squares are there in a 5×5 chessboard? - A) 55 - B) 65 - C) 75 - D) 85Answer: 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
3. Count rectangles in the grid:
**Question:** How many rectangles can be formed in a 3×4 grid? - A) 60 - B) 70 - C) 80 - D) 90Answer: 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
4. 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) 12Answer: C) 10 Solution: 4 small triangles + 3 medium triangles + 2 large triangles + 1 largest triangle = 10 Shortcut: Count systematically by size Concept: Triangle Counting
5. Count embedded circles:
**Question:** How many circles are completely hidden inside squares in the given figure? - A) 3 - B) 4 - C) 5 - D) 6Answer: B) 4 Solution: Look for circles whose entire boundary is within square boundaries Shortcut: Trace each circle’s boundary mentally Concept: Embedded Figures
6. 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) 12Answer: B) 8 Solution: The overlapping creates 8 distinct triangular regions Shortcut: Draw and shade different regions Concept: Overlapping Figures
7. 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) 18Answer: C) 17 Solution: 50 ÷ 3 = 16 complete cycles + 1 extra figure (triangle) Shortcut: Divide total by pattern length Concept: Pattern Recognition
8. 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) 5Answer: 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
9. 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) 6Answer: C) 4 Solution: 360° ÷ 90° = 4, so 4 figures will match after rotation Shortcut: Divide 360 by rotation angle Concept: Rotation Counting
10. Count triangles in star figure:
**Question:** A 5-pointed star (pentagram) has how many triangles? - A) 5 - B) 10 - C) 15 - D) 20Answer: B) 10 Solution: 5 small triangles + 5 larger triangles = 10 Shortcut: Count points and intersections Concept: Complex Figure Counting
11. 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) 30Answer: B) 21 Solution: 1 (large) + 4 (medium) + 16 (small) = 21 Shortcut: Sum of geometric progression Concept: Nested Figures
12. 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) 16Answer: B) 12 Solution: 3 × 4 = 12 hexagons Shortcut: Simple multiplication for regular patterns Concept: Pattern Counting
13. 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) 36Answer: A) 18 Solution: C(3,2) × C(4,2) = 3 × 6 = 18 Shortcut: Same as rectangle counting Concept: Parallelogram Counting
14. 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) 7Answer: B) 4 Solution: 3 pairwise intersections + 1 common to all three Shortcut: Draw Venn diagram mentally Concept: Overlapping Figures
15. 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) 19Answer: 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
