1
Regular polygons: each angle and the axes of symmetry
Mathematics Prep 1 Term 1, Unit 4, Lessons 5–6
GoalFind each angle of a regular polygon and count its axes of symmetry.
New words
regular polygon · a polygon with all sides equal and all angles equalinterior angle · an angle inside the polygon at a vertexaxis of symmetry · a line that folds the figure onto itselfshared side · a side that belongs to two shapes at once
Explanation
In a regular polygon all the angles are equal, so each interior angle is the angle sum divided by n: (n − 2) × 180° ÷ n. This gives 60° for a triangle, 90° for a square, 108° for a pentagon, 120° for a hexagon and 135° for an octagon. Be careful to divide the whole sum by n, not to stop at the sum. A regular polygon with n sides has n axes of symmetry. An irregular polygon may have fewer axes or none: a rectangle that is not a square has 2, a parallelogram that is not a rhombus has none.
| Regular polygon | Sides n | Angle sum | Each angle |
|---|---|---|---|
| equilateral triangle | 3 | 180° | 60° |
| square | 4 | 360° | 90° |
| pentagon | 5 | 540° | 108° |
| hexagon | 6 | 720° | 120° |
| octagon | 8 | 1080° | 135° |
Worked examples
Each angle. A regular pentagon: the angle sum is (5 − 2) × 180° = 540°. There are 5 equal angles, so each is 540° ÷ 5 = 108°.
The number of sides. Each angle of a regular polygon is 140°. Then (n − 2) × 180 = 140n, so 180n − 360 = 140n, 40n = 360 and n = 9. Check: (9 − 2) × 180 ÷ 9 = 1260 ÷ 9 = 140.
Axes of symmetry. A square has 4 axes: 2 through opposite vertices and 2 through the midpoints of opposite sides. An equilateral triangle has 3 axes and a regular hexagon has 6.
Two polygons. A regular hexagon and a square share the side PQ. At Q the hexagon has 120° and the square has 90°. The angles round a point add up to 360°, so x = 360° − 120° − 90° = 150°.
Class activity
Fold and count. With a grown-up, cut regular shapes from paper (triangle, square, hexagon) or use tiles. Fold each so it matches itself and count the different folds. Do you get n folds for n sides?
Practice
1
Find each angle of a regular octagon.
(8 − 2) × 180° = 1080°, and 1080° ÷ 8 = 135°.
2
Find each angle of a regular decagon (10 sides).
(10 − 2) × 180° = 1440°, and 1440° ÷ 10 = 144°.
3
Each angle of a regular polygon is 150°. How many sides has it?
(n − 2) × 180 = 150n, so 180n − 360 = 150n, 30n = 360 and n = 12.
4
How many axes of symmetry has a regular heptagon (7 sides)? A rectangle that is not a square? A scalene triangle?
7 axes. 2 axes. None.
5
Adel says each angle of a regular hexagon is 720°. What is wrong?
720° is the sum of all six angles. Each angle is 720° ÷ 6 = 120°.