Unit 15 Section 3 : Symmetry

In this section we revise the symmetry of objects and examine the symmetry of regular polygons.

There are two types of symmetry: reflective symmetry and rotational symmetry.

Reflective Symmetry
An object has reflective symmetry if it can be reflected in a particular line and looks the same as the original.
The line the object is reflected across is called a line of symmetry or a mirror line.

Look for lines of symmetry in the two shapes below:

In (a) there are two lines of symmetry, one horizontal and one vertical.
In (b) there are also two lines of symmetry, both diagonal.

Here are the shapes with the lines of symmetry drawn on:

Rotational Symmetry
An object has rotational symmetry if it looks the same as it did originally when rotated.

The order of rotational symmetry is the number of times an object looks the same as it did originally when it is rotated through 360°. Even if a shape appears to have no rotational symmetry then the order of rotational symmetry will still be 1, because every shape looks the same at the end of a 360° rotation as it did originally.

There is a centre of rotation about which the rotational symmetry occurs. There can only be one centre of rotation in a shape.

The three diagrams below show shapes with different orders of rotational symmetry.
One has an order of rotational symmetry of 1, one has order 2, and the other one has order 4.
Click on the shapes to see them rotate and count how many times they look the same in one turn.
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Note how they all rotate around a cross - this is the center of rotation.

Symmetries in regular polygons
Look at the regular heptagon below. A heptagon is a shape with seven sides and this one has equal sides and equal angles.
You can see that there are seven lines of symmetry, and the regular heptagon also has rotational symmetry order seven.

The order of rotational symmetry and the number of lines of symmetry of any regular polygon is equal to the number of sides.

 

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the Click this button to see if you are correct button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on Click on this button to see the correct answer to see the answer.

NOTE: In the next question you need to plot points and draw lines on grids.

To plot a point, just click on the grid with your left mouse button. To draw a line, hold down
the left mouse button at one point on the line and drag the pointer to another point on the line.
When you let go of the button the line will appear and it will automatically cross the whole grid.

If you make a mistake, press the delete key and the grid will be cleared.

Question 1
For each of the six diagrams below, draw on all the lines of symmetry and mark the centre of rotation point.
Once you have checked these with the button, fill in the order of rotational symmetry for each one and check that too.

(a)
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Order of rotational symmetry =
Order of rotational symmetry =
Order of rotational symmetry =
Order of rotational symmetry =
Order of rotational symmetry =
Order of rotational symmetry =
(b)
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(c)
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(d)
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(e)
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(f)
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Question 2
State the order of rotational symmetry and the number of lines of symmetry for each of the following shapes:

(a) Order of rotational symmetry =
Number of lines of symmetry =
(b) Order of rotational symmetry =
Number of lines of symmetry =
(c) Order of rotational symmetry =
Number of lines of symmetry =
(d) Order of rotational symmetry =
Number of lines of symmetry =
(e) Order of rotational symmetry =
Number of lines of symmetry =
(f) Order of rotational symmetry =
Number of lines of symmetry =

Question 3
Describe the symmetries of the shapes shown below:

(a) Order of rotational symmetry =
Number of lines of symmetry =
(b) Order of rotational symmetry =
Number of lines of symmetry =

Question 4
Describe the symmetry properties of each of the following triangles:

(a) Order of rotational symmetry =
Number of lines of symmetry =
(b) Order of rotational symmetry =
Number of lines of symmetry =
(c) Order of rotational symmetry =
Number of lines of symmetry =

Question 5
Complete the following table:

Shape Order of
rotational
symmetry		 Number of
lines of
symmetry		 Check your answers
Equilateral triangle
Square
Regular pentagon
Regular hexagon
Regular heptagon (7 sides)
Regular octagon
Regular nonagon (9 sides)
Regular decagon
Regular dodecagon (12 sides)

You have now completed Unit 15 Section 3
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