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.
