Lesson Video: Order of Rotational Symmetry | Nagwa Lesson Video: Order of Rotational Symmetry | Nagwa

# Lesson Video: Order of Rotational Symmetry Mathematics

In this video, we will learn how to find the order of rotational symmetry of a geometric figure and its angle of rotation.

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### 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.

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