Polygons · Ages 13–14
Regular polygons
All sides equal and all angles equal; n lines of symmetry, order n.
How to do it
Method
All sides equal and all angles equal; n lines of symmetry, order n
Regular hexagon: 6 lines of symmetry, a fit every 360° ÷ 6 = 60°
Another way
From the centre
Angle at the centre = 360° ÷ n: an octagon's 360° ÷ 8 = 45°
In full
A polygon is regular when all its sides are equal and all its angles are equal. One alone isn't enough: a rhombus has four equal sides but its angles needn't be equal; a rectangle has four equal angles but its sides needn't be. A regular n-sided polygon has n lines of symmetry and rotational symmetry of order n: every turn of 360° ÷ n brings it back onto itself. One way: count the lines. When n is odd, each passes through a vertex and the middle of the opposite side; when n is even, half pass through two opposite vertices and half through the middles of two opposite sides. Another: look from the centre. Joining the centre to the vertices makes n equal angles at the centre, each 360° ÷ n, and turning by one of them fits the shape onto itself. A regular hexagon: 6 lines of symmetry, rotational symmetry of order 6, a fit every 60°.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The picture shows a regular polygon. It has lines of symmetry.
Show the answer and working
Answer: 5
A regular polygon has as many lines of symmetry as sides: 5 sides, so 5 lines.
Example 2
A regular 9-sided polygon turns about its centre. The smallest turn that fits it back onto itself is °.
Show the answer and working
Answer: 40
Its 9 equal angles at the centre make a full turn: 360° ÷ 9 = 40°.
Example 3
All the sides of a 6-sided polygon are equal. It must be a regular 6-sided polygon.
True or false?
Show the answer and working
Answer: False
A triangle with equal sides has three angles of 60°, so it must be regular; with more than 3 sides, the sides can be equal while the angles aren't (a rhombus, say), so it needn't be.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A regular 6-sided polygon turns about its centre. The smallest turn that fits it back onto itself is °.
Answer:
2.The picture shows a regular polygon. It has lines of symmetry.
Answer:
3.All the sides of a 7-sided polygon are equal. It must be a regular 7-sided polygon.
True or false?
Answer:
4.O is the centre of the regular polygon. x =
Answer:
5.A regular polygon turned 60° about its centre fits onto itself for the first time. It has sides.
Answer:
All the answers
- 1. 60
- 2. 9
- 3. False
- 4. 72°
- 5. 6
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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