Study guides › Level 7 · Geometry

Counting diagonals

Level 7 · Geometry · for questions like “How many diagonals does a decagon have?”

💡 The idea

From each corner, you can draw a diagonal to every other corner except itself and its two neighbors: n − 3 diagonals. Doing that from all n corners counts each diagonal twice.

diagonals = n × (n − 3) ÷ 2

✏️ Worked examples

1. Decagon (10 sides): 10 × 7 ÷ 2 = 35.

2. A polygon with 20 diagonals: you need n(n − 3) = 40, and 8 × 5 = 40, so it has 8 sides.

⚠️ Watch out

• Halve at the end: each diagonal has two ends.

🎯 Try these

1. Hexagon (6 sides)
2. 12 sides
3. A polygon with 27 diagonals: how many sides?

Answers: 1) 9 2) 54 3) 9

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.