Video Transcript
In this video, we will learn how to
find the order of rotational symmetry of a geometric figure and how to find its
angle of rotation. Let’s begin by thinking about what
we mean by the order of rotational symmetry. The order of rotational symmetry of
a geometric figure is the number of times you can rotate the figure so it still
looks the same as the original figure.
Let’s have a look at a few examples
of what we mean by that. Let’s say we take this shape that
looks like an arrowhead and start to rotate it. After a 90 degrees’ clockwise
rotation, the arrow would look like this. Does it still look the same as the
original figure? No, it doesn’t. Let’s try another rotation. After a 180-degree rotation of the
original shape, the arrow will be pointing downwards. Does it still look the same? No, it doesn’t. After another 90 degrees’ turn, the
arrowhead is pointing to the left, and it still doesn’t look the same. It’s not until we rotate through
the full 360 degrees until we get the shape to look the same as it did at the
start. We can therefore say that this
figure has an order of rotational symmetry of order one. That’s because it only fits upon
itself once in a 360-degree turn.
Let’s have a look at a different
shape. Let’s take this rectangle and
rotate it 90 degrees clockwise. When we’ve rotated it, it doesn’t
look the same as the original figure. However, when the original shape is
rotated by 180 degrees, the image looks the same as the original figure. If we highlight a vertex of the
rectangle, when we rotated it 90 degrees, that vertex would appear on the top
right. And when we rotated it 180 degrees,
it would appear on the bottom right. The rectangle is effectively upside
down, but the image after the rotation still looks the same as the original
figure. The original shape rotated 270
degrees would look like this. And finally, a full 360-degree
rotation would bring us back to the starting point.
To find the order of rotational
symmetry then, we count the number of times the figure looked the same after
rotation. So the first time was after 180
degrees, and the second time was after 360 degrees, when it was back upon the
original starting position. Therefore, the order of rotational
symmetry of this rectangle is order two. And all rectangles will also have
an order of two.
So can we think of any shapes that
might have an order four? Let’s see if this square would have
an order of four. Let’s highlight a vertex of this
square and then try a rotation of 90 degrees. The vertex on the top left would
now be on the top right in the image. And the image of this square still
looks the same as in the original figure. So it’s already fitting upon itself
once. If we turn through another 90
degrees, the highlighted vertex is now on the bottom right. And it’s fitted upon itself
twice. After another 90-degree rotation,
it’s fitting upon itself three times. After a final 90-degree rotation,
it’s fitting upon itself four times, and it’s back in its original position. And so this square, like all
squares, has an order of rotational symmetry of four.
Before we look at some questions,
there’s one last important piece of terminology. Instead of being asked to give the
order of rotational symmetry of a shape, we might be asked if a shape has rotational
symmetry. We say that a shape has rotational
symmetry if the shape appears unchanged after rotation about a point by an angle of
rotation whose measure is strictly between zero degrees and 360 degrees. Zero degrees and 360 degrees are
excluded as that would be the original starting position.
Let’s look at the square and see if
by the definition it has rotational symmetry. Well, we know that after rotations
of 90 degrees, 180 degrees, and 270 degrees, this square appeared unchanged. It also did after 360-degree
rotation. But we don’t count that when we’re
thinking about rotational symmetry. So this square has got rotational
symmetry. When we consider the rectangle, the
rectangle also appeared unchanged after a rotation of 180 degrees. The rectangle also has rotational
symmetry.
So what about our first figure, the
arrowhead? The only time that the arrowhead
looked the same was when it had done the complete 360-degree rotation to be the same
as the original figure. However, we know from the
definition that a rotation of 360 degrees doesn’t count. So therefore, this arrowhead has no
rotational symmetry. Any shape that has an order of
rotational symmetry of order one has no rotational symmetry, and vice versa.
Let’s now have a look at some
questions where we find the order of rotational symmetry.
The following figure is an
equilateral triangle. Determine the order of rotational
symmetry of the figure. Option (A) order two, option (B)
order three, option (C) order four, option (D) order six, option (E) the figure does
not have rotational symmetry.
Let’s start by recalling that the
order of rotational symmetry of a geometric figure is the number of times you can
rotate the figure so it still looks the same as the original figure. And that’s over a rotation of 360
degrees. So let’s start by highlighting one
of the vertices on this triangle. And then we consider we turn or
rotate this triangle.
Because this is an equilateral
triangle, if we rotated this about the center, then after 120 degrees, the
highlighted vertex would now be at the base. The image after this rotation would
look the same as the original figure. If we then rotated the image
another 120 degrees or considered it as the original shape rotated 240 degrees, then
the highlighted vertex would now be on the bottom left. So the image looks like the
original shape. After a further 120 degrees of the
last image or a complete 360-degree rotation, then the shape would be back to the
original starting point.
To find the order of rotation, we
need to count how many times in this 360-degree rotation the image looked the same
as the original shape. So the image looked like itself
once after a 120-degree rotation, twice after 240 degrees, and finally a third time
when it was back in the original position. So we can give the answer that the
order of rotational symmetry is order three.
But before we finish with this
question, there’s a few things to note. The only reason that this triangle
has an order of rotational symmetry of three is because it was an equilateral
triangle. If we imagine we had even an
isosceles triangle and we rotated it through 360 degrees to see how many times it
looked the same as the original figure, we would find that that only happened
once. And that’s when it’s in the
original starting position. In this case, we would say that
this isosceles triangle would have on order of rotational symmetry of order one.
Notice that that’s also what we
mean when we say that the figure doesn’t have rotational symmetry, like we had in
option (E). However, we can give our answer
here as option (B) that this equilateral triangle and any equilateral triangle will
have an order of rotational symmetry of order three.
Let’s have a look at another
question.
Determine whether the following
statement is true or false. If a figure has one vertical line
of symmetry, then it also has rotational symmetry.
In this question, we need to recall
two different types of symmetry: firstly line symmetry, demonstrated through
reflection, and secondly rotational symmetry. We say that a shape has rotational
symmetry if the shape appears unchanged after a rotation about a point by an angle
whose measure is strictly between zero degrees and 360 degrees. That essentially means if we turn a
shape through 360 degrees and the shape after rotation looks the same as it did when
we started, then it has rotational symmetry.
Notice that we don’t include the
angle of zero degrees or 360 degrees as that would just be the starting
position. Perhaps the best way to answer a
question like this is to try drawing a few shapes that do have a line of symmetry
and see if they do or don’t also have rotational symmetry. So let’s pick a rectangle to begin
with.
The shape in question has to have a
vertical line of symmetry, and the rectangle does. Let’s then imagine that we have a
center of rotation in the center of the rectangle. And we begin to rotate the
rectangle. After 90 degrees, the rectangle
would look like this in green. But it doesn’t look the same as it
did to start off with.
Let’s continue to turn the shape
through another 90 degrees. Now, we can see that the rotation
sits on top of itself. The rectangle is effectively upside
down. But the original shape would look
the same as this rotated image. So the shape we started with did
have a vertical line of symmetry. And because it fits upon itself
after 180 degrees, we say that it does have rotational symmetry. This would be a good example to
show that the statement is true.
But let’s see if we can disprove it
and show it’s false. Let’s take this figure. It’s actually an isosceles
trapezoid because it’s got a pair of parallel sides and the two nonparallel sides
are equal in length. And that gives us this vertical
line of symmetry. Let’s consider what happens if we
rotate this shape through up to 360 degrees. Well, there are no points where
this trapezoid will look the same other than the original starting position. Even, for example, after a
180-degree rotation, the trapezoid would look upside down. That is, the base, which is longer,
will be at the top of the figure, which means that this shape would not have
rotational symmetry.
Considering the statement in the
question then — if a figure has one vertical line of symmetry, then it also has
rotational symmetry — we can give the answer as false. We found one occasion where it was
true, but we found on occasion where it’s false. Therefore, this statement will not
always be true.
In the next example, we’ll consider
a similar question. Only this time we’ll see what
happens when there are two lines of symmetry.
Determine whether the following
statement is true or false. If a figure has one horizontal and
one vertical line of symmetry, then it also has rotational symmetry.
To consider what we mean by both of
these types of symmetry, let’s take an example. Here we have a square. A square has one horizontal and one
vertical line of symmetry. It does have some others as well,
but let’s just consider these two. We say that a shape has rotational
symmetry if the shape appears unchanged after rotation by an angle strictly between
zero degrees and 360 degrees. In other words, if we cut out this
square, for example, if we had it made out of cookie dough and we cut it out with a
cookie cutter, and we then turned the square through 90 degrees, it would still fit
into the original shape. Because it does that after an angle
of rotation of 90 degrees or 180 degrees, rather than just in its original position,
then this square has rotational symmetry.
So for the square that we’ve drawn
then, the statement is true. It has got the horizontal and
vertical lines of symmetry, and it has rotational symmetry. Let’s see if we can create an
interesting geometric figure based on the lines of symmetry and see what happens
with the rotational symmetry. Let’s take our two lines of
symmetry and draw a shape in one of these corners.
Because of the horizontal line of
symmetry, our shape would look like this. And then let’s consider the
vertical line of symmetry, which gives us the completed shape. If we rotated this figure through
an angle of 180 degrees, it would fit upon itself. We know that it would be upside
down, but it appears unchanged after rotation by 180 degrees. Therefore, it has rotational
symmetry.
We can use this pictorial example
to show that for any shape that has a horizontal and a vertical line of symmetry,
then it must also have rotational symmetry. So the answer would be true. Be careful, however. If a figure just has one line of
symmetry, then we can’t say that it also has rotational symmetry.
Before we look at the final
question, we need to cover another important concept, the angle of rotation. The angle of rotation is the
smallest angle for which the figure can be rotated to coincide with itself. Let’s recap a few geometric figures
and their orders of rotational symmetry.
A rectangle has an order of
rotational symmetry of order two, an equilateral triangle of order three, and a
square of order four. If we’re looking for the smallest
angle for which the shape can be rotated and fit upon itself, for the rectangle,
that would be 180 degrees, the equilateral triangle 120 degrees, and the square
would be 90 degrees. You may have noticed that there is
a relationship between the angle of rotation and the order of rotational
symmetry. The angle of rotation is equal to
360 degrees divided by the order of rotational symmetry. For the rectangle, 360 degrees
divided by two give us 180 degrees. For the equilateral triangle, 360
degrees divided by three gives us 120 degrees. And for the square, 360 degrees
divided by four would give us 90 degrees. Let’s see how we can apply this in
the next question.
Does the following figure have
rotational symmetry? If yes, find the angle of
rotation.
In order to answer this question,
let’s recall what we mean by the order of rotational symmetry. The order of rotational symmetry of
a geometric figure is the number of times you can rotate the figure so it still
looks the same as the original figure. And that’s within a 360-degree
rotation. So let’s consider the figure that
we’re given. It is in fact a square, as both the
length and the width would be the same length. If we count the squares along both
sides, then each of these would be six squares long.
When we’re working out rotational
symmetry, we’ll need to consider any picture or pattern on the figure. This figure would in fact have two
lines of symmetry along the diagonals. And while we’re not thinking about
lines of symmetry, this will help us when we’re rotating. If we rotate this square through 90
degrees, the lengths would fit along like the original shape, but the patterns
wouldn’t match. This white square, for example, can
sit on a black square and still look the same as the original image.
So let’s instead think about a
180-degree rotation of the original shape. We would now have a white square on
top of a white square and black squares on top of black squares. Therefore, after a rotation of 180
degrees, the rotated image would look the same as the original shape. This would happen again after a
complete rotation of 360 degrees. Because the shape looks the same as
the original figure once after a rotation of 180 degrees and then for a second time
at 360 degrees, we say that the order of rotational symmetry is two. Any shape that has an order of
rotational symmetry more than one has got rotational symmetry. This means that our answer would be
yes.
So next, we need to find the angle
of rotation. The angle of rotation is the
smallest angle for which the figure can be rotated to coincide with itself. It can be calculated using the
formula that the angle of rotation is equal to 360 degrees divided by the order of
rotational symmetry. So we need to divide 360 degrees by
two, giving us a value of 180 degrees. We can therefore give our answer
that yes, this figure has rotational symmetry and the angle of rotation is 180
degrees.
We can now summarize some of the
key points of this video. Firstly, we saw that the order of
rotational symmetry of a geometric figure is the number of times you can rotate the
figure over 360 degrees so it still looks the same. A shape has rotational symmetry if
the shape appears unchanged after a rotation about a point by an angle whose measure
is strictly between zero degrees and 360 degrees. Finally, we saw that to calculate
the angle of rotation, we divide 360 degrees by the order of rotational
symmetry.
