Counting diagonals of a polygon.
- (i) How should we define a diagonal of an n-gon? How many diagonals does an n-gon have? Make a table for small values of n. A 3-gon has no diagonals. A 4-gon has 2. How many diagonals does a 5-gon have? A 6-gon? Try to find a pattern and guess the answers for n = 7 and n = 8. Check your guesses by systematic counting.
- (ii) Can you guess a formula for the number of diagonals? How many diagonals get added when we increase the number of sides by 1? Can you now justify why the formula you guessed is true for all n?
Answer: A diagonal joins two vertices that are not adjacent. Diagonals for n = 3, 4, 5, 6, 7, 8: 0, 2, 5, 9, 14, 20. Formula: n(n − 3)2. Going from n to n + 1 sides adds n − 1 diagonals.
Step-by-step solution
Idea: Each vertex is joined by a diagonal to every vertex except itself and its two neighbours, i.e. to n − 3 vertices. Counting from every vertex counts each diagonal twice.
(i) How should we define a diagonal of an n-gon? How many diagonals does an n-gon have? Make a table for small values of n. A 3-gon has no diagonals. A 4-gon has 2. How many diagonals does a 5-gon have? A 6-gon? Try to find a pattern and guess the answers for n = 7 and n = 8. Check your guesses by systematic counting.
- Definition: a diagonal of an n-gon is a segment joining two vertices that are not adjacent (not the two ends of a side).½ mark
- Counting (see the picture): n = 3: 0; 4: 2; 5: 5; 6: 9. Differences 2, 3, 4 suggest the next differences 5, 6: guess 7-gon: 14, 8-gon: 20.1 mark
- Systematic check for n = 7: from vertex 1 draw diagonals to 3, 4, 5, 6 (4 of them); from 2 to 4, 5, 6, 7 (4); from 3 to 5, 6, 7 (3); from 4 to 6, 7 (2); from 5 to 7 (1): 4 + 4 + 3 + 2 + 1 = 14 ✓. For n = 8: 5 + 5 + 4 + 3 + 2 + 1 = 20 ✓.½ mark
Diagonal = segment joining two non-adjacent vertices. n = 3, 4, 5, 6, 7, 8 gives 0, 2, 5, 9, 14, 20 diagonals.
(ii) Can you guess a formula for the number of diagonals? How many diagonals get added when we increase the number of sides by 1? Can you now justify why the formula you guessed is true for all n?
- Formula: each of the n vertices has n − 3 diagonals (all vertices except itself and its two neighbours). That gives n(n − 3) ends of diagonals; each diagonal has two ends, so the number is n(n − 3)2. Check: n = 7 gives 7 × 42 = 14 ✓.1 mark
- Adding a side: put a new vertex V between neighbours U and W of an n-gon. V gets diagonals to the n − 2 old vertices other than U and W, and the old side UW becomes a diagonal: n − 2 + 1 = n − 1 new diagonals. Check: (n + 1)(n − 2)2 − n(n − 3)2 = n − 1 ✓. Starting from 0 diagonals for n = 3 and adding 2, 3, 4, … step by step gives the formula for every n.1 mark
Number of diagonals = n(n − 3)2; going from n to n + 1 sides adds n − 1 diagonals.
A diagonal joins two non-adjacent vertices; an n-gon has n(n − 3)/2 diagonals (0, 2, 5, 9, 14, 20 for n = 3 to 8), and adding one side adds n − 1 diagonals.
Answer to write in the exam
(i)
Diagonal: segment joining two non-adjacent vertices
n: 3, 4, 5, 6, 7, 8
diagonals: 0, 2, 5, 9, 14, 20
Check n = 7: 4 + 4 + 3 + 2 + 1 = 14; n = 8: 5 + 5 + 4 + 3 + 2 + 1 = 20
(ii)
Each vertex: n − 3 diagonals; n vertices; each diagonal counted twice
⇒ number of diagonals = n(n − 3)/2
New vertex between U and W: n − 2 new diagonals + UW becomes a diagonal ⇒ n − 1 added
(n + 1)(n − 2)/2 − n(n − 3)/2 = n − 1 ✓
Common mistakes that cost marks
- Forgetting to divide by 2, so each diagonal is counted twice (getting 28 instead of 14 for a 7-gon).
- Counting n − 1 or n − 2 diagonals per vertex; a vertex is not joined by a diagonal to itself or to its two neighbours.
- When adding a vertex, forgetting that the old side UW becomes a diagonal.
How this can come in the exam
MCQ (1 mark)
The number of diagonals of a 10-sided polygon is
- 20
- 35
- 45
- 70
Show answer
(B) 35
10 × 72 = 35.
Short answer (2 marks)
A polygon has 27 diagonals. How many sides does it have?
Show answer
n(n − 3)2 = 27 ⇒ n2 − 3n − 54 = 0 ⇒ (n − 9)(n + 6) = 0 (1 mark) ⇒ n = 9 sides (1 mark).Try one yourself
How many diagonals does a 12-gon have, and how many more does a 13-gon have?
Show answer
12 × 92 = 54; a 13-gon has 13 × 102 = 65, i.e. 11 more (= 12 − 1).
More questions like this
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