A twenty-sided shape has 170 diagonals. Drawing them is not a method, and the count that gets there is one subtraction away from a question a child has already answered.
Join every pair of the twenty corners and you get 190 lines. Twenty of those are the sides the shape already has, so 170 are diagonals. That is the whole argument, and it is worth more than the formula it is equivalent to — n(n−3)/2 gives the same number and explains nothing.
Why this sheet works
Two wrong answers are worth reaching. 190 is the pair count with the sides left in, and a child who says it has done the hard half right. 340 is the subtraction done without the halving, which is the mistake the pair-counting sheet exists to teach and which comes back the moment the subtraction takes the attention.
Count them without drawing them. Show your working.
110 people sit in a ring. Everyone shakes hands with everyone else except the two sitting next to them.How many handshakes are there?Why
214 towns are arranged in a ring, and a road already joins each town to the two beside it. New roads are built so that every pair of towns is joined directly.How many new roads are built?Why
Diagonal countingUp to 20 corners
Name: Date:
Count them without drawing them. Show your working.
3A polygon has 12 sides.How many diagonals does it have?Why
413 people sit in a ring. Everyone shakes hands with everyone else except the two sitting next to them.How many handshakes are there?Why
Diagonal counting · Up to 20 cornersAnswer key
1. Every pair of the 10 could be joined. Each person reaches 9 others, so 10 × 9 = 90 — and each pair is counted from both ends, so 90 ÷ 2 = 45 lines in all. But 10 of them join neighbors, so take them off: 45 − 10 = 35 handshakes. More than one way works here and it never counts the neighbor pairs: each person joins all but itself and its two neighbors, so 10 × 7 = 70, and 70 ÷ 2 = 35.2. Every pair of the 14 could be joined. Each town reaches 13 others, so 14 × 13 = 182 — and each pair is counted from both ends, so 182 ÷ 2 = 91 lines in all. But 14 of those roads already exist, so take them off: 91 − 14 = 77 new roads. More than one way works here and it never counts the roads already there: each town joins all but itself and its two neighbors, so 14 × 11 = 154, and 154 ÷ 2 = 77.3. Every pair of the 12 could be joined. Each corner reaches 11 others, so 12 × 11 = 132 — and each pair is counted from both ends, so 132 ÷ 2 = 66 lines in all. But 12 of them are its sides, so take them off: 66 − 12 = 54 diagonals. More than one way works here and it never counts the sides: each corner joins all but itself and its two neighbors, so 12 × 9 = 108, and 108 ÷ 2 = 54.4. Every pair of the 13 could be joined. Each person reaches 12 others, so 13 × 12 = 156 — and each pair is counted from both ends, so 156 ÷ 2 = 78 lines in all. But 13 of them join neighbors, so take them off: 78 − 13 = 65 handshakes. More than one way works here and it never counts the neighbor pairs: each person joins all but itself and its two neighbors, so 13 × 10 = 130, and 130 ÷ 2 = 65.
