Video: Symmetry in 3D Shapes

In this video, we will learn how to determine whether a 3D shape has plane symmetry, axis symmetry, or neither and state the number of planes or axes of symmetry it has.

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Video Transcript

In this video, we will learn how to identify if a three-dimensional shape has plane symmetry or axis symmetry. We’ll also learn how to calculate the number of planes or axes of symmetry. We can begin by thinking about plane symmetry. But before we do that, it may be worthwhile recapping some symmetry in a two-dimensional shape. Let’s take this square and see if we can find any symmetry. Well, we could draw a line of reflection or a line of symmetry like this and horizontally like this or, indeed, diagonally like these two lines. Each of these lines would create a mirror image on the other side.

So, taking this concept of a line of reflection and translating it into a three-dimensional shape means that the line becomes a plane. We can describe a plane of symmetry as a two-dimensional surface that cuts a solid into mirrored or congruent halves. If we imagine our orange rectangle or plane here to be something sharp that will cut our prism into two congruent halves, then we could create this plane of symmetry here on this cube or prism. The front section of the prism would be congruent and a mirror image of the section of the prism at the back. So, we found one plane of symmetry in this prism. And as we go through this video, we’ll see other planes of symmetry that could be found in a prism such as this.

Next, let’s take a look at axis symmetry, which may often also be referred to as rotation symmetry. We also see rotation symmetry in two-dimensional shapes. If we rotated this two-dimensional shape about the point through 360 degrees, we would see that after half a turn, that the object would fit onto itself again. And then completing another half turn through to 360 degrees, we would find that the object fitted back onto its original shape. So, taking the concept of a point for rotation and translating it into three dimensions means that we’re looking for a line about which a three-dimensional shape can be rotated. We can define an axis of symmetry as a line in space about which an object may be rotated through 360 degrees and repeat. In other words, it will fit onto itself.

If we take this prism to be formed from a regular or an equilateral triangle, then as we rotated this around the axis, it would fit on itself once, twice, and then three times on the original starting point. The term “the order of rotational symmetry” means the number of times an object fits onto itself during a 360-degree rotation. So, the order of rotational symmetry for our triangular prism would be three. There are a few things that we could say about this prism. We could say the prism has an axis of symmetry, or we could refer to this by saying the prism has axis symmetry. We can now look at some questions on plane and axis symmetry.

How many planes of symmetry does this solid have?

Let’s begin by recalling that a plane of symmetry is a two-dimensional surface that cuts a solid into two mirrored or congruent halves. Let’s imagine we have this pink rectangle or plane cutting down through the solid. We could then create a plane of symmetry like this. These two portions of the solid at the front and the back would be mirror images. And they would be congruent so long as the lengths at the front are equal to the lengths at the back. Let’s see if we can find any other planes of symmetry on the solid. This time, let’s imagine our plane looking like this and cutting downwards through the solid. This would create two congruent mirrored halves. And so, we found another plane of symmetry. Let’s see if there are any more.

Let’s say we created a plane that looks like this, would it be a plane of symmetry? Well, in this case, we would have created two congruent halves. However, these halves are not a mirror image of each other. If we compare this to the two-dimensional equivalent where we find a line of symmetry. In a rectangle, we do not have a line of symmetry that looks like this since we don’t have mirror images on both sides of the line. The plane that we have drawn here in this prism would only work as a plane of symmetry if we know that the prism is a cube. So, let’s remove this diagram and see if we can find any more planes of symmetry.

If we take a look at our first diagram, we could see that we have divided the width of our prism. In our second prism, we divided the lengths into congruent pieces. So, how about trying to divide the height of this prism into two congruent pieces? Our plane might then look something like this. If the height is divided into two equal pieces, then we would have created two mirrored congruent halves. So here, we found three different planes of symmetry for this solid. There are no other planes of symmetry, so this will be our final answer.

Let’s have a look at another question on plane symmetry.

Does a square pyramid have plane symmetry?

In this question, we need to recall what a square pyramid is and, also, what plane symmetry is. We can start by drawing a square pyramid. That’s simply a pyramid with a square on the base. We can recall that a plane of symmetry is a two-dimensional surface that cuts a solid into two mirrored and congruent halves. If a solid has a plane of symmetry, then we can describe it as having plane symmetry. So, let’s see if we can create a plane of symmetry in this square pyramid. Let’s imagine this pink rectangle or plane cutting downwards through the pyramid. When this plane cuts down through the pyramid, creating equal lengths on the front and the back, then we would have created mirrored and congruent halves. And so, we found a plane of symmetry.

A plane in this direction would work for any square pyramid even, for example, in a really tall square pyramid like this. We could still create the plane of symmetry. So, we have answered the question, does a square pyramid have plane symmetry, but let’s see if we can quickly find any other planes of symmetry. In our first example, we divided this length into two pieces. So, let’s see if we can create another plane of symmetry by dividing this length into two pieces. And so, we could create two congruent mirrored pieces, giving us another plane of symmetry. Even though these two planes of symmetry look very similar, they are in fact two distinct planes of symmetry.

We can also create two more planes of symmetry where the planes divide the vertices of the base. The first one would look like this and the next one would divide the other vertices. Now, we have established that there are four planes of symmetry in a square pyramid. Showing just one of these would be sufficient to say that, yes, a square pyramid does have plane symmetry.

In the next question, we’ll look at axes of symmetry.

The following solid has an axis of symmetry about the shown axis. What is the order of rotational symmetry about the shown axis?

We can recall that an axis of symmetry is a line in space about which an object may be rotated through 360 degrees and repeat. In other words, through this 360-degree rotation, the shape will fit on itself more than once. The order of rotational symmetry tells us how many times that will happen. We can mark one of the lower vertices in pink. Beginning a rotation of this around the axis of symmetry, then after 90 degrees of a rotation, this pink vertex would appear here. After another 90-degree turn, the pink vertex would be at the top. Another 90 degrees places this pink vertex here. And completing our 360-degree rotation would place this pink vertex back where it started.

We can therefore say that this shape has fitted onto itself or repeated four times, which means that the answer for the order of rotational symmetry about the shown axis is four.

In the final question, we’ll look at both the plane symmetry and the axis symmetry of a regular prism.

Consider the following regular prism. How many planes of symmetry does the prism have? Does the prism have an axis of symmetry?

Here, we’re told that the prism is regular, which means that it’s formed from a regular polygon. And so, the six sides on this polygon will all be the same length. As a six-sided polygon is a hexagon, here we would have a hexagonal prism. Let’s look at the first question regarding planes of symmetry. A plane of symmetry is a two-dimensional plane which divides this 3D object into two mirrored congruent halves. Let’s imagine that we’ve drawn a line of symmetry between the midpoints of two opposite sides on the hexagon. We could then create a two-dimensional plane by extending along the length of this prism. And so, we have found one plane of symmetry as we’ve created two mirrored congruent halves of the prism.

Let’s see if we can find another plane of symmetry. In the same way as before, we could create a line of symmetry joining the midpoints of opposite lengths. We could then create a two-dimensional plane by extending along the length of the prism. We can find another plane of symmetry by joining the midpoints of the other two opposite sides and create a plane of symmetry like this. It might be tempting to think that because we have a hexagon, that there must be six planes of symmetry joining the midpoints. But in fact, we only have three like this as each of the lines joining the opposite points only counts as one plane. So, so far, we have found three planes of symmetry. Let’s clear these and see if we can find any more.

Our last approach was to join the midpoints of opposite lengths, but this time let’s try joining opposite vertices. A line of symmetry on the two-dimensional hexagon would be like this. And therefore, a plane of symmetry would be like this, remembering that we can say that it’s a plane of symmetry as we’ve created two congruent mirrored halves. We might then predict that we could create another two planes of symmetry. Joining opposite vertices in our hexagon and then extending along the prism, which give us these other two planes of symmetry. And therefore, so far, we have found six planes of symmetry. But are there any other ones? Let’s consider the prism.

So far, we have found the midpoints of opposite lengths and joined those. We then joined the opposite vertices. But what about if we cut the prism along its length? We can see that we would have two congruent mirrored halves. And therefore, we have found another plane of symmetry. We have therefore found a total of seven planes of symmetry. And as there are no other ones, then this would be our answer.

There’s a handy tip for working out the planes of symmetry in a regular prism like this. And that is if we consider the two-dimensional polygon at the base of the prism, then the number of planes of symmetry will be equal to the number of lines of symmetry plus one. In our example, the hexagon has six lines of symmetry and the final plus one bit always comes from this last plane of symmetry which cuts along the prism length. But notice this does only work for regular prisms.

We can now look at the final part of this question, does the prism have an axis of symmetry? We can recall that an axis of symmetry is a line in space about which a shape can be rotated and repeat or fit upon itself. If we consider the hexagon by itself and we were to create a point at the center of it. Then, the hexagon would have rotational symmetry about this point. We could then extend this point to be a line going through the prism. We can now think if we rotated this prism around this axis, would it repeat?

If we marked one of the vertices with an orange dot and then as we rotate this shape, the vertex in orange would be in this position. We could then continue our rotation, and so turning this orange vertex around further. Continuing the rotation through 360 degrees, we would find that this orange dot fits upon itself. Therefore, we can see how the entire prism would repeat or fit upon itself during a 360-degree rotation. We weren’t asked to give the order of a rotational symmetry. That’s the number of times the shape fits upon itself. But if we were, we could say that the order of rotational symmetry here is six.

We were asked if the prism has an axis of symmetry, and we’ve demonstrated that it does repeat as it turns through 360 degrees. And so, our answer for the final part is yes.

We can now summarize some of the key points of this video. We saw that a plane of symmetry is a two-dimensional surface that cuts a solid into two mirrored congruent halves. An axis of symmetry is a line about which an object may be rotated and repeat. The order of rotational symmetry is the number of times an object repeats when rotated 360 degrees around the axis of symmetry. And finally, as with many topics involving three-dimensional shapes, neat, clear diagrams are very useful to help us find the axes and planes of symmetry.

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