Polygons • Topic 3 of 3

Diagonals

A diagonal joins two non-adjacent vertices. From each of the n vertices you can draw a line to (n−3) others — every vertex except itself and its two neighbours — giving n(n−3) endpoints, but each diagonal is counted from both ends, so divide by 2: number of diagonals = n(n−3)/2. A quadrilateral has 4(1)/2 = 2, a pentagon 5(2)/2 = 5, a hexagon 6(3)/2 = 9, an octagon 8(5)/2 = 20. CAT sometimes runs this backwards — "a polygon has 35 diagonals, how many sides?" — so be ready to solve n(n−3)/2 = D for n, which is a quick quadratic or, faster, a guess-and-check near √(2D). The area-of-a-regular-polygon idea also lives here: a regular n-gon splits into n identical isosceles triangles from its centre, the backbone of the (1/2)×perimeter×apothem area formula.

✅ Solved examples

1. How many diagonals does a regular octagon (8 sides) have?
n(n − 3)/2 = 8 × 5 / 2 = 40/2 = 20 diagonals.
2. A polygon has 35 diagonals. Find the number of sides.
n(n − 3)/2 = 35 ⇒ n(n − 3) = 70 = 10 × 7 ⇒ n = 10. (A decagon.)
3. How many more diagonals does a decagon have than a hexagon?
Decagon: 10 × 7 / 2 = 35. Hexagon: 6 × 3 / 2 = 9. Difference = 35 − 9 = 26.
4. Find the area of a regular hexagon of side 6 cm.
Area = (3√3/2) × a² = (3√3/2) × 36 = 54√3 ≈ 93.5 cm². (It is six equilateral triangles of side 6.)

✏️ Practice — try these, take hints as needed

1. How many diagonals does a polygon with 12 sides have?
Use n(n − 3)/2.
n − 3 = 9.
12 × 9 / 2.
54
2. A polygon has 20 diagonals. How many sides does it have?
n(n − 3)/2 = 20.
n(n − 3) = 40.
Find two factors differing by 3.
8 sides
3. How many diagonals can be drawn from a single vertex of a 15-gon?
From one vertex you skip itself and 2 neighbours.
Count = n − 3.
15 − 3.
12
4. A regular hexagon has side 10 cm. Find its area.
Area = (3√3/2) a².
a² = 100.
(3√3/2) × 100.
150√3 ≈ 259.8 cm²
5. How many diagonals does a polygon have if it has 9 sides?
n(n − 3)/2.
n − 3 = 6.
9 × 6 / 2.
27

📝 Topic test — 8 questions

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