Counting Figures — Triangles, Squares, Rectangles

🎯 Learning Objective

Count figures in standard grids using direct formulas, and count size-by-size when the figure is non-standard.

💡 Concept

• Squares in an n×n grid = n(n+1)(2n+1)/6 — it is the sum 1² + 2² + ... + n²
• Check with n = 2: 4 small + 1 big = 5, and formula gives 2×3×5/6 = 5 — matches
• Rectangles in an m×n grid: a rectangle = choose 2 vertical lines AND 2 horizontal lines → [m(m+1)/2] × [n(n+1)/2]
• Triangles in a triangle with each side divided into n parts (standard grid) = n(n+2)(2n+1)/8 — drop the fraction for odd n
• Check with n = 2: 4 small + 1 big = 5, and formula gives 2×4×5/8 = 5 — matches
• Non-standard figure? Count by SIZE: all 1-unit shapes, then 2-unit, then bigger — never jump randomly

🧮 Key Formulas

Squares in n×n grid = n(n+1)(2n+1)/6

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Rectangles in m×n grid = [m(m+1)/2] × [n(n+1)/2]

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Triangles (side cut into n parts) = n(n+2)(2n+1)/8 (drop fraction)

✏️ Easy Example

Q. How many squares are there in a 2×2 grid?

1. Formula: n(n+1)(2n+1)/6 with n = 2
2. = 2 × 3 × 5 / 6 = 5
3. Check: 4 small + 1 big = 5

Answer: 5

🇮🇳 Real-Life Example

A chessboard has 204 squares in total (8×9×17/6), not 64 — and 1296 rectangles. This exact chessboard question has appeared in railway papers.

📝 Exam-Level Example

Q. How many squares are there in a 3×3 grid?

1. n = 3: 3 × 4 × 7 / 6
2. = 84 / 6 = 14
3. Check: 9 (1×1) + 4 (2×2) + 1 (3×3) = 14

Answer: 14

📝 Exam-Level Example

Q. A big triangle has each side divided into 4 equal parts with the standard grid drawn. How many triangles in total?

1. n = 4: 4 × 6 × 9 / 8
2. = 216 / 8 = 27

Answer: 27

🪄 Memory Trick

Rectangles in a square grid = [n(n+1)/2]². For the chessboard: (8×9/2)² = 36² = 1296. And remember — every square IS a rectangle, so the rectangle count already includes squares.

⚠️ Common Mistakes

• ❌ Answering 64 for chessboard squares — bigger squares (2×2 up to 8×8) also count
• ❌ Using the triangle formula on figures that are not the standard subdivided triangle
• ❌ Counting the same figure twice when counting manually — always fix a size order

🏆 Exam Tips

• ✅ Verify any formula on a tiny case first (2×2 grid = 5 squares) before trusting it in the exam
• ✅ For odd n in the triangle formula, the .125 fraction is simply dropped: n = 3 → 105/8 → 13

📌 Summary

• Squares: n(n+1)(2n+1)/6; chessboard = 204
• Rectangles: pick 2 + 2 lines → [m(m+1)/2][n(n+1)/2]
• Standard triangle grid: n(n+2)(2n+1)/8, drop fraction
• Non-standard figures: count strictly size by size