Numbers in Circles & Triangles

🎯 Learning Objective

Find the missing number when values sit at the corners and centre of circles or triangles.

💡 Concept

• The outer numbers (corners or edges) combine to make the centre number
• Try in order: sum of outer numbers, then product, then twice the sum, then sum of squares
• Two complete figures are given as 'training' — the rule must fit BOTH
• Triangles: the three corner numbers build the middle number
• Circles: the numbers around the rim build the centre number
• Once the rule fits both complete figures, apply it to the incomplete one

🧮 Key Formulas

Centre = sum of outer, or product of outer, or 2 × sum, or sum of squares

✏️ Easy Example

Q. Three triangles have corner numbers (5, 6, 7), (4, 8, 3), (9, 2, 6) and centres 18, 15, ?. Find the missing centre.

1. Try sum of corners: 5 + 6 + 7 = 18 ✓
2. Confirm: 4 + 8 + 3 = 15 ✓
3. Apply: 9 + 2 + 6

Answer: 17

🇮🇳 Real-Life Example

A cricket scoreboard adds three batsmen's runs into a partnership total — corner numbers feeding a centre total is that same everyday addition.

📝 Exam-Level Example

Q. Three triangles have corners (4, 2, 3), (5, 3, 2), (6, 2, 4) and centres 24, 30, ?. Find the missing centre.

1. Sum gives 9, not 24 → try product
2. 4 × 2 × 3 = 24 ✓, 5 × 3 × 2 = 30 ✓
3. Apply: 6 × 2 × 4

Answer: 48

📝 Exam-Level Example

Q. Three circles have rim numbers (1, 2, 3), (2, 2, 3), (3, 1, 4) and centres 12, 14, ?. Find the missing centre.

1. Sum is 6 but centre is 12 → centre = 2 × sum
2. 2 × (1+2+3) = 12 ✓, 2 × (2+2+3) = 14 ✓
3. Apply: 2 × (3+1+4) = 2 × 8

Answer: 16

🪄 Memory Trick

Compare the centre with the plain sum first. Centre bigger than sum → try product or 2×sum; centre near the sum → it probably IS the sum with a small tweak.

⚠️ Common Mistakes

• ❌ Fixing a rule from just one figure — always verify on the second complete figure
• ❌ Forgetting to try product when the centre is much larger than the sum
• ❌ Reading the corner numbers in the wrong order (rotate the figure to a fixed starting corner)

🏆 Exam Tips

• ✅ Keep a mental menu: sum, product, 2×sum, sum of squares — try them in that order
• ✅ Write the sum and product of the first figure's outer numbers; one of them usually matches the centre

📌 Summary

• Outer numbers build the centre number
• Try sum → product → 2×sum → sum of squares
• Rule must fit both complete figures
• Centre ≫ sum hints at product