In this video, we will learn exactly how to recognize whether a 3D shape has aircraft symmetry, axis symmetry, or neither and also state the variety of planes or axes of symmetry that has.

You are watching: How many planes of symmetry does a rectangular prism have

### Video Transcript

In this video, we will learn just how to identify if a three-dimensional shape has plane symmetry or axis symmetry. We’ll likewise learn how to calculation the number of planes or axes the symmetry. We can begin by thinking around plane symmetry. But prior to we carry out that, it may be worthwhile recapping part symmetry in a two-dimensional shape. Let’s take this square and see if we can find any type of symmetry. Well, we might draw a line of reflection or a heat of symmetry like this and horizontally favor this or, indeed, diagonally favor these 2 lines. Every of this lines would create a mirror picture on the various other side.

So, taking this principle of a heat of reflection and translating it right into a three-dimensional shape way that the line becomes a plane. We can explain a plane of symmetry together a two-dimensional surface that cut a solid right into mirrored or congruent halves. If us imagine our orange rectangle or airplane here to it is in something sharp the will reduced our prism into two congruent halves, then we can create this plane of symmetry right here on this cube or prism. The front ar of the prism would certainly be congruent and also a mirror image of the ar of the prism at the back. So, we uncovered one plane of the opposite in this prism. And as us go through this video, we’ll see other planes of the contrary that can be found in a prism such together this.

Next, let’s take it a look at axis symmetry, which might often likewise be described as rotation symmetry. We also see rotation the contrary in two-dimensional shapes. If we rotated this two-dimensional shape around the suggest through 360 degrees, we would view that after half a turn, that the object would certainly fit onto chin again. And also then perfect another half turn with to 360 degrees, us would find that the thing fitted back onto its original shape. So, acquisition the ide of a point for rotation and also translating it into three dimensions way that we’re looking for a line about which a three-dimensional shape have the right to be rotated. Us can specify an axis the symmetry as a line in an are about which an object may be rotated with 360 degrees and repeat. In various other words, it will certainly fit ~ above itself.

If we take this prism come be developed from a continual or an it is provided triangle, then as we rotated this around the axis, it would fit on itself once, twice, and then three times ~ above the original beginning point. The ax “the stimulate of rotational symmetry” method the variety of times an item fits top top itself during a 360-degree rotation. So, the stimulate of rotational symmetry for our triangular prism would be three. There are a couple of things the we could say around this prism. We could say the prism has actually an axis of symmetry, or we might refer to this by saying the prism has axis symmetry. We can now look in ~ some inquiries on airplane and axis symmetry.

How many planes of symmetry go this heavy have?

Let’s start by recalling that a aircraft of the contrary is a two-dimensional surface ar that cut a solid into two copy or congruent halves. Let’s imagine we have actually this pink rectangle or aircraft cutting down v the solid. We might then produce a plane of symmetry choose this. These two parts of the solid in ~ the front and the ago would be winter images. And also they would certainly be congruent so lengthy as the lengths in ~ the front room equal to the lengths at the back. Let’s watch if we deserve to find any kind of other plane of symmetry on the solid. This time, stop imagine our airplane looking like this and cutting downwards with the solid. This would create two congruent copy halves. And so, we discovered another aircraft of symmetry. Let’s check out if there are any type of more.

stop say we developed a aircraft that looks prefer this, would certainly it be a aircraft of symmetry? Well, in this case, we would have developed two congruent halves. However, these halves space not a mirror photo of each other. If us compare this come the two-dimensional identical where we uncover a line of symmetry. In a rectangle, we execute not have a heat of symmetry that looks prefer this because we don’t have mirror photos on both sides of the line. The airplane that us have drawn here in this prism would only job-related as a airplane of symmetry if we understand that the prism is a cube. So, let’s eliminate this diagram and see if we can discover any much more planes that symmetry.

If us take a look in ~ our first diagram, we might see that us have split the broad of our prism. In our second prism, we separated the lengths right into congruent pieces. So, how about trying to division the elevation of this prism into two congruent pieces? Our airplane might climate look something choose this. If the height is divided into 2 equal pieces, then we would have produced two copy congruent halves. Therefore here, we discovered three various planes the symmetry for this solid. There are no other planes of symmetry, for this reason this will be our last answer.

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

go a square pyramid have aircraft symmetry?

In this question, we should recall what a square pyramid is and, also, what plane symmetry is. We have the right to start by drawing a square pyramid. That’s just a pyramid with a square top top the base. We have the right to recall the a plane of the opposite is a two-dimensional surface ar that cuts a solid right into two mirrored and congruent halves. If a solid has actually a aircraft of symmetry, then us can define it as having airplane symmetry. So, let’s watch if we can create a airplane of the contrary in this square pyramid. Let’s imagine this pink rectangle or aircraft cutting downwards through the pyramid. When this airplane cuts down with the pyramid, developing equal lengths on the front and the back, then we would have created mirrored and also congruent halves. And also so, we found a aircraft of symmetry.

A airplane in this direction would job-related for any type of square pyramid even, because that example, in a really tall square pyramid like this. We might still create the plane of symmetry. So, we have actually answered the question, go a square pyramid have aircraft symmetry, but let’s check out if we can conveniently find any kind of other plane of symmetry. In our very first example, we split this length into two pieces. So, let’s view if us can produce another plane of the contrary by dividing this length right into two pieces. And so, we might create 2 congruent copy pieces, providing us another airplane of symmetry. Also though these 2 planes of the opposite look really similar, they room in fact two distinct planes the symmetry.

us can also create two much more planes of symmetry whereby the planes division the vertices of the base. The an initial one would certainly look like this and also the next one would certainly divide the other vertices. Now, us have developed that over there are 4 planes of the opposite in a square pyramid. Showing simply one that these would certainly be sufficient to speak that, yes, a square pyramid walk have aircraft symmetry.

In the next question, fine look in ~ axes of symmetry.

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

We have the right to recall that an axis of symmetry is a heat in room about which things may be rotated through 360 degrees and repeat. In other words, v this 360-degree rotation, the form will to the right on itself much more than once. The order of rotational the opposite tells us how countless times that will certainly happen. We have the right to mark one of the lower vertices in pink. Beginning a rotation the this roughly the axis the symmetry, then after 90 levels of a rotation, this pink peak would show up here. After an additional 90-degree turn, the pink vertex would be in ~ the top. Another 90 degrees places this pink vertex here. And completing our 360-degree rotation would location this pink vertex earlier where it started.

we can because of this say the this shape has actually fitted onto itself or repeated 4 times, which means that the answer for the bespeak of rotational symmetry around the shown axis is four.

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

consider the following consistent prism. How many planes that symmetry go the prism have? go the prism have an axis the symmetry?

Here, we’re told that the prism is regular, which means that it’s created from a continuous polygon. And also so, the 6 sides ~ above this polygon will certainly all it is in the very same length. As a six-sided polygon is a hexagon, right here we would have a hexagonal prism. Stop look at the an initial question regarding planes the symmetry. A aircraft of the opposite is a two-dimensional aircraft which divides this 3D object into two mirrored congruent halves. Let’s imagine the we’ve drawn a heat of symmetry in between the midpoints of 2 opposite sides on the hexagon. We might then create a two-dimensional airplane by extending along the size of this prism. And so, us have found one plane of symmetry together we’ve produced two mirrored congruent halves of the prism.

Let’s check out if we can uncover another aircraft of symmetry. In the same way as before, we can create a heat of symmetry joining the midpoints of the opposite lengths. We could then develop a two-dimensional plane by prolonging along the length of the prism. Us can find another airplane of the contrary by involvement the midpoints of the various other two the opposite sides and create a airplane of symmetry favor this. It might be tempting come think that since we have actually a hexagon, that there should be 6 planes of the opposite joining the midpoints. But in fact, us only have three like this as each that the present joining the contrary points just counts together one plane. So, for this reason far, us have uncovered three airplane of symmetry. Let’s clear these and also see if we have the right to find any type of more.

our last technique was to join the midpoints of the contrary lengths, however this time let’s try joining the opposite vertices. A line of the opposite on the two-dimensional hexagon would certainly be choose this. And also therefore, a airplane of symmetry would be favor this, remembering the we deserve to say the it’s a aircraft of symmetry together we’ve developed two congruent copy halves. We might then predict the we can create another two airplane of symmetry. Joining opposite vertices in ours hexagon and also then prolonging along the prism, which provide us these other two airplane of symmetry. And therefore, for this reason far, we have found six plane of symmetry. However are there any kind of other ones? Let’s think about the prism.

therefore far, we have discovered the midpoints of opposite lengths and joined those. Us then joined the the opposite vertices. Yet what about if we cut the prism follow me its length? We can see that us would have actually two congruent copy halves. And also therefore, we have discovered another plane of symmetry. Us have as such found a total of 7 planes the symmetry. And as there are no various other ones, then this would be ours answer.

yes sir a handy tip for functioning out the airplane of symmetry in a continuous prism prefer this. And also that is if we take into consideration the two-dimensional polygon at the base of the prism, climate the number of planes the symmetry will be same to the number of lines the symmetry to add one. In our example, the hexagon has six currently of symmetry and also the last plus one bit constantly comes from this last aircraft of symmetry which cut along the prism length. But an alert this does only occupational for continual prisms.

We can now look in ~ the final part of this question, go the prism have an axis that symmetry? We deserve to recall that an axis of the opposite is a heat in room about i m sorry a shape deserve to be rotated and repeat or fit upon itself. If we think about the hexagon by itself and we were to produce a allude at the center of it. Then, the hexagon would have rotational symmetry around this point. We might then prolong this point to be a line going with the prism. We deserve to now think if us rotated this prism roughly this axis, would certainly it repeat?

If we significant one the the vertices with an orange dot and also then together we turn this shape, the peak in orange would certainly be in this position. We could then proceed our rotation, and so turning this orange vertex around further. Proceeding the rotation with 360 degrees, us would uncover that this orange dot fits top top itself. Therefore, we can see just how the whole prism would certainly repeat or fit top top itself throughout a 360-degree rotation. Us weren’t asked to provide the stimulate of a rotational symmetry. It is the variety of times the shape fits upon itself. Yet if we were, we might say that the bespeak of rotational symmetry here is six.

we were request if the prism has actually an axis the symmetry, and we’ve demonstrated that it walk repeat together it turns through 360 degrees. And also so, ours answer for the final component is yes.

We can now summarize some of the an essential points of this video. We experienced that a plane of the contrary is a two-dimensional surface that cut a solid into two copy congruent halves. One axis of the contrary is a line around which things may be rotated and repeat. The bespeak of rotational symmetry is the variety of times an object repeats as soon as rotated 360 degrees approximately the axis that symmetry.

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And finally, similar to many topics entailing three-dimensional shapes, neat, clean diagrams are an extremely useful to help us discover the axes and also planes of symmetry.