Polygons
The names of shapes from triangle to decagon, their sides, their lines of symmetry, their diagonals and their perimeter.
A polygon is a flat shape with straight sides that join up all the way round, with no gaps. Triangles, squares and the six-sided cells of a honeycomb are all polygons. A circle is not, because it has no straight sides. The word comes from Greek and means "many corners". Polygons are everywhere: the stop sign in many countries has eight sides, and a classic soccer ball is sewn from pentagons and hexagons. Once you know their names, you can describe almost any flat shape you see.
The names of polygons
Each name tells you the number of sides. Most names start with an old Greek or Latin word for a number, and many of those words are still hidden in everyday English.
| Sides | Name | How to remember it |
|---|---|---|
| 3 | triangle | tri means three, as in a tricycle |
| 4 | quadrilateral | quad means four, as in a quad bike |
| 5 | pentagon | penta means five |
| 6 | hexagon | hexa means six; hexagon and six both have an x |
| 7 | heptagon | hepta means seven |
| 8 | octagon | octo means eight, like the eight arms of an octopus |
| 9 | nonagon | nona means nine |
| 10 | decagon | deca means ten, as in a decade (ten years) |
| 12 | dodecagon | dodeca means twelve |
A shape has as many corners as it has sides. A polygon is called regular when all its sides are the same length and all its angles are the same size. A square is a regular quadrilateral. A rectangle that is not a square is not regular, because two of its sides are longer than the other two.
Example 1
What is the name of this shape?
- Put your finger on one corner.
- Move round the shape and count each side: 1, 2, 3, 4, 5, 6, 7.
- Stop when you are back where you started. There are 7 sides.
- A shape with 7 sides is a heptagon.
Answer: A heptagon
Lines of symmetry
A line of symmetry is a line that cuts a shape into two halves that match exactly. If you fold the shape along the line, one half lies perfectly on top of the other. It is also called a mirror line: put a mirror on it, and the half you see plus its picture in the mirror make the whole shape.
Remember
A regular polygon has as many lines of symmetry as it has sides.
So an equilateral triangle has 3 lines, a square has 4 and a regular hexagon has 6. Where do the lines go?
- When the number of sides is odd (3, 5, 7 ...), every line goes from a corner to the middle of the opposite side.
- When the number of sides is even (4, 6, 8 ...), half of the lines join two opposite corners, and the other half join the middles of two opposite sides.
Example 2
How many lines of symmetry does a regular hexagon have?
- A hexagon has 6 corners, in 3 pairs of opposite corners. That gives 3 lines.
- It has 6 sides, in 3 pairs of opposite sides. The lines through their middles give 3 more.
- 3 + 3 = 6.
Answer: 6 lines of symmetry
Example 3
How many lines of symmetry does a rectangle (not a square) have?
- Fold it from left to right: the halves match. That is 1 line.
- Fold it from top to bottom: the halves match. That is 2 lines.
- Fold it along a diagonal: the corners do not land on each other. That is not a line of symmetry.
Answer: 2 lines of symmetry
Diagonals
A diagonal is a straight line that joins two corners that are not next to each other. The sides of the shape are not diagonals. A triangle has no diagonals at all, because every corner is next to the other two. A square has 2.
To count the diagonals of a bigger shape, start at one corner. It can join every other corner except itself and its two neighbours, because the lines to the neighbours are sides. So from each corner there are 3 fewer diagonals than there are corners.
- From one corner, count the diagonals: the number of sides minus 3.
- Multiply by the number of corners.
- Divide by 2, because every diagonal has two ends and was counted from both of them.
Remember
Number of diagonals = (n(n − 3))/2
n is the number of sides.
Example 4
How many diagonals does a pentagon have?
- From each corner: 5 − 3 = 2 diagonals.
- For all 5 corners: 5 × 2 = 10.
- Each one was counted twice: 10 ÷ 2 = 5.
Answer: 5 diagonals
Example 5
How many diagonals does a heptagon have?
- From each corner: 7 − 3 = 4 diagonals.
- For all 7 corners: 7 × 4 = 28.
- Each one was counted twice: 28 ÷ 2 = 14.
Answer: 14 diagonals
| Shape | Sides | Diagonals |
|---|---|---|
| quadrilateral | 4 | 2 |
| pentagon | 5 | 5 |
| hexagon | 6 | 9 |
| heptagon | 7 | 14 |
| octagon | 8 | 20 |
Perimeter
The perimeter of a shape is the total length all the way round its edge. To find it, add up the lengths of all the sides. When the shape is regular, all the sides are the same, so you can multiply instead of adding.
Remember
Perimeter of a regular polygon = number of sides × length of one side
Example 6
All the sides of this hexagon are 7. What is its perimeter?
- A hexagon has 6 sides.
- Each side is 7, so the perimeter is 6 × 7 = 42.
- Check by adding: 7 + 7 + 7 + 7 + 7 + 7 = 42.
Answer: 42
Example 7
A regular octagon has a perimeter of 48. How long is each side?
- An octagon has 8 equal sides.
- Share the perimeter between them: 48 ÷ 8 = 6.
- Check: 8 × 6 = 48.
Answer: 6
Try it yourself
1.How many sides does a nonagon have?
Show the answer
9
Nona means nine. A nonagon has 9 sides and 9 corners.
2.What is the name of this shape?
Show the answer
A decagon
It has 10 sides, and deca means ten.
3.How many lines of symmetry does a regular pentagon have?
Show the answer
5
A regular polygon has as many lines of symmetry as sides. Each line goes from a corner to the middle of the opposite side.
4.All the sides of a heptagon are 6. What is its perimeter?
Show the answer
42
A heptagon has 7 sides, and 7 × 6 = 42.
5.A regular hexagon has a perimeter of 54. How long is each side?
Show the answer
9
A hexagon has 6 equal sides, and 54 ÷ 6 = 9.
6.How many diagonals does a decagon have?
Show the answer
35
From each corner, 10 − 3 = 7 diagonals. 10 × 7 = 70, and each one is counted twice, so 70 ÷ 2 = 35.
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