Lesson video

Welcome, everyone.

I'm Mr. Gratton, and in today's maths lesson, we will delve deep into understanding what defines a prism, how to name prisms and to look at types of prism that you may not even be aware exist.

A prism is a 3D shape, a polyhedron with base that is a polygon and a parallel opposite face that is identical.

The corresponding edges of these two polygons are joined by parallelograms. A cross section is a 2D face made from cutting straight through any plane of any 3D object, and a cross section of a prism that is made parallel to the base will also be congruent to that base.

Pause here to familiarise yourself with some of the other keywords that we'll be covering during this lesson.

First up, let's have a look at the basics when defining a prism, but before that, which of these 3D shapes is a prism? Pause here to consider using any prior knowledge that you may have.

Only b and e are prisms. The others are not.

Let's have a look as to why that's the case.

For a 3D shape to be a prism, any cross sections taken parallel to the base must be exactly the same uniform two-dimensional shape to any other cross section taken parallel to that first cross section.

The left-hand shape shows this.

This hexagonal cross section is uniform throughout the entire shape.

However, for this shape, the polygon at the top of this 3D shape is much smaller than the polygon at the bottom.

As these parallel cross sectional shapes are not the same size as each other, this is definitely not a prism.

We can see this in a different way by overlaying the two 3D shapes on top of each other.

The cross sectional hexagon at the bottom of the prism is congruent in size to the hexagon at the top.

However, the cross sectional hexagon of the non-prism has become smaller than these other two.

It is important to note that only cross sections parallel to the base have to be congruent.

Two cross sections that are not parallel do not need to be congruent with each other.

For this rightmost shape, these two hexagons are not congruent because they are not parallel.

Notice how the top cross section is at a distinctly different angle to the base or the top face of this prism.

However, this rightmost shape is definitely still a prism because the parallel cross sections are still congruent even though these non-parallel cross sections are not.

It is also important to note that only cross sections parallel to the base that are taken in one direction have to be congruent to each other.

For example, in this right-hand prism, these vertical cross sections at one of the faces and through the centre of the prism are different in size.

This is okay as it is not this set of cross sections that define it as a prism.

Rather it is the horizontal ones taken at the base, as seen in that leftmost shape, that define it as a prism.

The 2D cross sectional shapes of a prism must be polygons.

For example, in this leftmost shape, the prism has equilateral triangles as its cross sectional polygon.

This cylinder on the right has congruent circles as its cross sectional shape, but because circles are not polygons, this is not a prism, even if it does share many other properties of a prism.

Furthermore, all other faces that are not the cross sectional ones on a prism must be parallelograms. For now, we will only look at right prisms, where those other faces are rectangles.

For this rightmost shape, these faces are trapezia, not parallelograms, and therefore this shape is not a prism.

As long as the cross sectional shape is the same congruent polygon all throughout, the polygon itself can have any property, any number of sides, any sized angles, or for it to be regular or irregular, and lastly, it is very important to be aware that to be a prism, the cross section must be uniform throughout the entire length of the shape.

It is not enough to just compare the two faces at the opposite ends of the shape, as you can see on this rightmost shape.

Whilst it has congruent rectangles at the very top and bottom of the shape, the rectangle at the centre is much smaller.

Therefore this is not a prism.

Okay, check number one.

This 3D shape is definitely a prism, as shown by these four 2D shapes.

Which of these words and phrases must be true to describe these four 2D shapes? Pause now to choose at least one correct word.

And all four of these 2D shapes must be parallel to each other, must all be congruent or identical to each other, and they must all be polygons.

Okay, on to check number two.

Which of these statements best describes why Izzy is incorrect? Pause here to look through these possible explanations.

B is correct.

The cross sections have not been taken in the correct direction.

Parallel cross sections of a prism definitely have to be congruent but only in one direction of the shape.

Izzy has not chosen the correct direction.

If two parallel cross sections are not congruent, the shape will still be a prism if you can find a different direction in which all of the polygonal cross sections are congruent.

Cross sections can be anywhere along a 3D shape, not just at the faces themselves.

Here are some familiar 3D shapes for the first question of this practise session.

Which of these 3D shapes match all of the properties of a prism? Pause here to look through all seven 3D shapes.

And for question number two, complete the sentences with the missing words provided at the bottom to give an explanation of what a prism is.

Pause now to look at each explanation, and be careful as each word can only be used once.

These diagrams all show the same prism.

Which of the diagrams shows the correct cross sections that help define it as a prism? Pause now to provide a thorough explanation.

Question number four.

Andeep says that this 3D shape is not a prism because the two cross sections he found are not congruent to each other.

Pause now to explain why Andeep is incorrect.

On to the answers.

C, e and f are all prisms. For question number two, a prism is a 3D object.

All of its 2D cross sections must be polygons, not circles, for example.

In at least one direction, all parallel cross sections must be congruent to each other.

For question number three, b is correct because the cross sections are all of the following: polygons, parallel to each other, and congruent.

A and c did not satisfy all three of these criteria, and for question number four, this might still be a prism.

Andeep has not taken two cross sections that are parallel to each other, and so this test is not effective at confirming whether a shape is a prism or not.

One cross section looks parallel to the top of the object, whilst the other cross section is at an angle compared to that bottom face.

Now we know what defines a prism, let's put some names to different types of prisms. Well, how? We take the face or cross section that is uniform throughout the entire prism and look at its shape and properties.

This one is a quadrilateral.

Specifically, it is a trapezium.

Therefore we can call it either a quadrilateral prism or a trapezium-based prism.

The name of a prism also specifies whether the cross sectional polygon is regular or not.

For example, square-based prism, this is a regular quadrilateral.

Rectangular prism and parallelogram-based prism.

This is also true for other types of polygons such as the regular hexagonal prism and the hexagonal prism or irregular hexagonal prism.

Okay, quick check.

Match the prism to its name.

Pause now to consider which 2D polygon lies on the base face of each prism.

And here are the correct answers.

Great work if you matched any of these correctly.

Prisms can be used in many different contexts, including construction.

Here are some common examples.

The names I-beam and L-frame are clearly named after the shape of their cross sectional base.

However, they can also still be described using their cross sectional polygon in more mathematical terms, but if the number of sides of the prism is tricky to see in a three-dimensional view, you can always sketch the two-dimensional cross sectional polygon first and count the number of sides from that sketch instead.

This I-beam is actually a dodecagon on a dodecagonal prism, and this L-frame is actually a hexagon on a hexagonal prism.

Okay, next check.

By first sketching the cross sectional polygon of this prism, count how many sides the cross sectional polygon actually has.

Pause now to give yourself some time to sketch the polygon at the base of this prism.

This polygon has eight sides, and pause here for the next check to consider what the most appropriate name for this prism would be.

It would be an octagonal prism because it has eight sides, which means the polygon is an octagon.

On to the second set of practise questions.

For question number one, place these 3D shapes into the correct row in the table.

Pause now to consider each shape.

For question number two, on squared paper, sketch the cross sectional polygon for each of these prisms. Pause the video here to try sketching all three.

As each is a sketch, do not worry about the accuracy of the proportions, as long as the shape is pretty much the same as the ones that you see on screen.

For question number three, write down a suitable name for each prism.

Pause the video here to try and name them all, and there may be more than one correct answer for each prism.

And the answers are a and f are triangular prisms, c and d are quadrilateral prisms, b is a pentagonal prism, and e, well, it isn't a prism at all.

For question two, here are some examples of what your sketches may have looked like, and for question three, pause here to have a look through some of the possible answers.

If your answer is different to any of those on screen, can you justify why it is still correct? And pause again here for this final prism.

Every type of prism that we've looked at so far is actually just a special case of a more general form of a prism.

Here's what I mean.

We've looked at right prisms because there is a right angle between the base polygon and the rectangular faces.

However, these type of prisms also exist, the oblique prism.

An oblique prism looks more slanted or leaning compared to a right prism.

In an oblique prism, the cross sectional polygons that define it as a prism are still congruent and parallel to each other throughout the entire length of the prism, as these are still important features that define what a prism is.

However, the non-cross sectional faces of an oblique prism are no longer perpendicular to the polygonal cross sections, like this angle here being less than 90 degrees rather than at a right angle.

Furthermore, these faces can be any type of parallelogram, sometimes a rectangle and sometimes not.

On to the final check.

Match the prism to its name.

Pause now to consider both the polygon at the base of the prism and whether the prism is right or just more generally oblique.

Here are the answers.

A is oblique because it looks more slanting, and the base face is a trapezium, therefore oblique trapezium prism.

For b, the base face is also a trapezium, but this time, it looks right rather than generally oblique.

For c, this one looks very oblique because it looks very slanted.

The base face is a triangle.

It may not look equilateral.

This may be down to the perspective with which we are seeing it from, and for d, we've got a right prism because it doesn't look slanted.

The base face is right angled and triangular, therefore its full name is right right-angled triangular prism, and on to the final set of practise questions.

Which of these are right prisms and which are oblique? Place each into the correct row of this table, and for question number two, for each prism, write down a suitable name.

Pause here to try both of these questions, and when naming a prism, be as detailed as possible.

For question number three, prisms a and b are the same prism seen from different perspectives.

Each of Aisha and Lucas are looking at this prism from these different perspectives.

Who is correct? And pause now to consider what each person is saying and why they may be saying it from the perspective they are looking through.

On to the answers.

B and c are right prisms, whilst a and d are generally oblique.

Pause here to have a look at some of the suggested names for these prisms, and again, can you justify the name if it isn't one of these on screen? And for question number three, both Aisha and Lucas are correct.

Some quadrilateral prisms can be both right and oblique at the same time.

If looking at the prism face on, at the base cross sectional face, it will appear like a right prism, whilst looking at it straight on at a different face will make it look oblique.

Well done on trying to wrap your head around some challenging 3D concepts in a lesson where we have identified a range of properties that define prisms from non-prisms. We've also identified and named different prisms by looking at the properties of the polygon at the cross sectional face of the prism.

Furthermore, we have broadened our horizons on what a prism is, from just being familiar with right prisms, to understanding that oblique prisms with general parallelogram faces adjacent to the base also exist.

Thank you so much for joining me today in today's lesson.

Have a brilliant rest of your day, and I hope to see you soon for another maths lesson.

Take care.