Lesson video

Hello, I'm Mrs. Lashley, and I'm gonna be working with you as we go through the lesson today.

I really hope you're looking forward to it and you're ready to try your best.

Today's learning outcome is to be able to calculate the surface area of a pyramid.

The screen here has got some new words that we'll be using during the lesson.

You may wish to pause the video and work through them, but we will be covering them during the lesson as well.

So our lesson on finding the surface area of a pyramid is going to be broken into two learning cycles.

The first learning cycle is to look at the properties of a pyramid, and then we're going to move on to the second learning cycle where we'll be focusing on the surface area of a pyramid.

So let's make a start at looking at those properties of a pyramid.

So all of these can be classified as 3D shapes.

Maybe you can name some of them.

I'm hoping you feel familiar with at least a couple.

Some of them are polyhedrons and some are not.

So have you used that word before polyhedrons? And so I've split them now into those that are polyhedrons and those that are not.

Can you see why they've been grouped like this? So pause the video and think about why they are one group and why the others are a different group.

What properties have they got in common? Well, polyhedrons need to have all polygonal faces.

So on the right hand side, the ones that are not, we've got a cylinder, which has circular faces.

We have a sphere, which has no flat face.

No flat surface.

It's one curved surface and a cone, which also has a circular face.

A circle is not a polygon.

So is this 3D shape a polyhedron? Pause the video and when you're ready to check your answer, press play.

So yes, this would be classified as a polyhedron.

All the faces are polygonal.

Some polyhedrons can then go further to be classified as prisms. A prism is a polyhedron with a base that is a polygon and a parallel opposite face that is identical.

The corresponding edges of the two polygons are joined by parallelograms. So let's have a look at prisms and not prisms. So here we've got prisms. They fit that definition.

The top one is a pentagonal prism and the bottom one is known as a cuboid, but we could think of it as a rectangular prism, whereas these two are not.

These two do not fit the definition of a prism.

There is another group of polyhedron that are not prisms, but instead are pyramids.

So let's think about pyramids against not pyramids.

And remember, our lesson today is about the surface area of pyramids.

So what is a pyramid? So here are examples of pyramids and here is a non-example, a polyhedron that is not a prism but is not also a pyramid.

So what does a 3D shape need to have to be classified as a pyramid? It needs to have triangular faces that are attached to a polygonal base.

So that base can be any polygon.

The triangular faces all meet at a point and that's known as the apex.

And the apex for the triangular-based pyramid is being labelled there.

So a pyramid is a shape that has got triangular faces that meet at an apex.

And those triangular faces are connected to, or attached to the polygonal base.

So here is a check.

Which of these 3D shapes are pyramids? Pause the video and then when you're ready to check, press play.

A, C and E are all pyramids.

B is a prism, D is a cone, and F is a composite 3D object.

A is a pyramid.

The view of that pyramid might have made it more difficult to see, but hopefully you can identify that you've got four triangular faces meeting at a point, which is the apex, and they've come off what looks to be a kite.

It might not be a kite, but some quadrilateral.

So the quadrilateral is the base and that is a polygon.

C is a pentagonal based pyramid.

And E is another quadrilateral potentially that is a rectangular-based pyramid.

So all four of these are square-based pyramids.

The base is polygonal a square and there are triangular faces that meet an apex.

So it is got the definition of being pyramid.

So all of those are square based.

Here are the nets for each of the square-based pyramids.

There is a link to a (indistinct) profile where you can have a look at these a little bit more interactive if you want to.

So you can click on that and explore that a little bit more.

So these are the nets for those four.

What are the similarities with the nets? So pause the video and think about that or maybe click on that link and have an exploration before you come back here.

Similarities, there are one square face and four triangular faces.

So the square face is the base of our pyramid and we know that we then have all triangular faces that are adjoined to the edges of that base, which in this case is the square and they will meet at an apex.

What are the differences with these nets? This is the type and sizes of the triangular faces.

So one of the nets, the bottom left one, all four of those triangles are congruent to each other.

They look to be isosceles.

We don't know that for certain, but they look to be isosceles.

Whereas the top right looks like we've got two pairs of congruent triangles.

All of them look like they're right angle triangles, but their size is slightly different.

So the difference means that the apex of each pyramid is in a different location.

So those triangles be in different sizes and different number of them will meet at an apex when they fold up 'cause a net will fold into its 3D shape and the apex will be in a different place.

So this is the plan view of the four square-based pyramids and it allows a location of the apex to be seen.

So it's the same four that you've seen.

So there's an apex right in the centre, there's an apex within the plan view within the square, but off centre.

We've got an apex that's out external to the base and we've got an apex on the vertex of one of the squares.

And we can classify these pyramids a little bit further.

So the one where the apex is above the centre of the base is a called a right pyramid.

The other three are all oblique pyramids.

So if the apex is directly above the centre of the base, regardless of what shape that base is, if it's above the centre, we would call it a right pyramid.

Anywhere else and it's an oblique pyramid.

There are links there which will allow you to explore the views if you're struggling to recognise the apex and the plan view.

Here's a check.

All square-based pyramids are the same.

true or false.

And then justify your answer with A or B.

Pause the video, when you're ready to check that press play.

So this is false and it's because the apex can be in various positions.

They're the same in terms of having one square face and four triangular faces, but they're not the same in terms of the shape of them.

So pyramids can be named by the shape of their polygonal base.

For example, the one on the left would be an oblique hexagonal-based pyramid.

Why is it oblique? Well, the apex is definitely not above the centre of that hexagon.

So oblique hexagonal based because the base is a hexagon and a pyramid because it fits the definition of our 3D shape.

That is a polyhedron where the base and triangular faces meet at an apex.

The right hand one, a right rectangular-based pyramid.

So the apex here is above the centre of the rectangle and the rectangle is its base shape and it's a pyramid again because we've got these triangular faces that meet an apex and a polygonal base.

So name the pyramid from this description.

A pyramid where the apex is directly above a vertex of the pentagonal base.

Pause the video and when you're ready to check your naming, press play.

Oblique pentagonal-based pyramid.

So it's oblique because the apex is directly above a vertex, it's not above the centre.

So for it to be accord a right pyramid, it needs to be above the centre.

Pentagonal base because we were told that the base was pentagonal and a pyramid because we were told it was a pyramid.

So we're onto the first task where we're looking at properties of a pyramid.

So question one, I'd like you to fill the blanks in this sentence.

Press pause and when you're ready for the next few questions, press play.

So the last three questions are on this slide.

So there's question two, sketch a right rectangular-based pyramid.

Question three, sketch a hexagonal-based pyramid.

And question four, explain why these are not pyramids.

Press pause, and when you press play, we'll go through our answers to task A.

Question one, you need to fill the blanks.

So a pyramid has a polygonal base and triangular faces that meets at an apex.

If the apex is directly above the centre of the base, then it is a right pyramid, otherwise it is an oblique pyramid.

How did you get on with that? So there's your definition of a pyramid and then the distinction between right and oblique.

Question two, you needed to sketch a right rectangular-based pyramid.

So here are a couple of examples of my sketches.

The base needs to be a rectangle and the apex must be directly above the centre of the rectangle.

And that's because we were told it was right.

So a rectangular based, so that's why it's a rectangle.

And then the right part means that the apex is directly above the centre.

Question three, you needed to sketch a hexagonal-based pyramid.

This time it didn't tell you whether it was right or oblique.

So that was up to you.

So I've got some examples here.

The main thing was that the base needs to be a hexagon and we probably always think of hexagons as a regular hexagon, but a hexagon is just a shape with straight edges and there needs to be six of them.

So there is one there that we might call an L shape, and that is a hexagonal-based pyramid where you place the apex, as I said, doesn't matter.

Question four, explaining why these were not pyramids.

So on part A, the base is not polygonal, it's a circle, it's a cone.

Part B, it was a prism.

So the triangular faces do not meet at an apex.

So it has triangular faces, but for a pyramid, all of the triangular faces will meet at an apex.

And for C, it has too many non-triangular faces.

So it is not a prism, it's not a pyramid.

It has triangular faces and it also has non-triangular faces.

On a pyramid, you might have a triangular-based pyramid and therefore all of the faces are triangles.

But at most you only have one non-triangular face on a pyramid.

Okay, so we've established what a pyramid is.

We are now gonna move on to the second learning cycle where we're gonna really focus on finding the surface area of a pyramid.

So in the first learning cycle we established that pyramids can be right pyramids or oblique pyramids.

The surface area will be calculated by finding the total area of all the faces, regardless of which type of pyramid is.

But in this learning cycle, all pyramids will be right pyramids, which means that the apex is directly above the centre of the polygonal base.

A square-based pyramid can be shown in various ways.

So here I've got a square-based pyramid.

It could be shown using plan and elevations.

So my plan view, my front and my side view.

And here you can see that this is a right pyramid and we know it's a right pyramid because from that plan view, the apex is above the centre.

So that's one way that we could show it.

Alternatively, we could use a net.

So we could unfold our 3D object flat to be a 2D representation.

And we call this the net.

The surface area of the square-based pyramid can be calculated by considering the different faces.

We need to add up the areas of each individual face to find the surface area.

So I've got my square, which is A, and then I've got four congruent triangles, which I've labelled all of those B.

The reason that they are congruent triangles is because it is a square and because it is a right pyramid.

So this area of the square is easy to find, that's 10 times 10, which is 100 square centimetres.

But what's the area of the triangles? Each triangle has the same area, but what is it? Well, we don't have the required information at this point, so we're gonna have to use other mathematical skills to get it.

So half times base times perpendicular height is the area of a triangle.

So we need to find that perpendicular heights because we know that these are isosceles triangles and we know that they are isosceles because it's a right pyramid, then I can use the symmetry of the triangle and that's the altitude to be the perpendicular height.

So I can use Pythagoras' theorem because I know that that perpendicular height is splitting the triangle into two congruent right angle triangles where the base would be five and the hypotonus would be 13.

So using Pythagoras' theorem, I can solve it and find that the perpendicular height of this isosceles triangle is 12.

Now I've got the required information to get the area.

So the area would be half times 10 times 12, which is 60 square centimetres.

So with a square the square base has an area of 100 and each triangular face has an area of 60.

So our surface area is the sum of those areas.

So 100 plus 60 plus 60 plus 60 plus 60.

Obviously you could do 60 times four instead and that's 340 square centimetres.

So here is a check for you.

Work out the surface area of this square-based pyramid using the areas given.

So pause the video and when you're ready to check your surface area, press play.

So your surface area was a case of adding 36 and four 54s together, which was 252 square centimetres.

A pyramid regardless of what pyramid, but a pyramid has many different lengths or heights.

So I've got my same square-based pyramid here.

So the square is 10 centimetres by 10 centimetres and that lateral length is 13.

The perpendicular height of the pyramid is the perpendicular length from the apex to the base.

So remember we were only working with the right pyramids in this learning cycle.

So this perpendicular height is the distance from the apex directly down, so perpendicular to the base.

The perpendicular height of the triangle is the perpendicular length from the apex to the edge of the base.

And we just calculated that to find the area earlier, which was 12 centimetres.

So we've got many different lengths or heights that you could be needing to work out or you could be given, and you need to work out which one is necessary for what you are doing.

Is the perpendicular height of the pyramid the same as the perpendicular height of the triangle? So just think about that for a moment.

Pause the video and when you finished, press plate and I'll carry on.

So for a right pyramid, the two perpendicular lengths are different.

If you have an oblique pyramid and only certain oblique pyramids, then there is cases where they are the same.

But for a right pyramid, which is what we are working with here, the two perpendicular lengths are different.

So the perpendicular height of the pyramid is different to the perpendicular height of the triangular face.

So here's a check, true or false, the perpendicular height of the pyramid and the perpendicular heights of the triangular face are equal and justify your answer.

Pause the video and then when you're ready to check that, press play.

So that's false.

For a right pyramid, the perpendicular height of the pyramid will be shorter than the perpendicular height of the triangle.

And you can probably recognise that because the triangle is going to be laying down slightly.

It's not sat upright when the pyramid is formed.

So it needs to be longer in order to reach the apex.

So the perpendicular height of this pyramid is five centimetres that's labelled there with the dash line.

So remember the perpendicular height of the pyramid is the distance from the apex to the base.

The surface area of this rectangular-based pyramid is the total of the rectangular face and the four triangular faces.

That's what surface area is, is the total of all of the areas of each face.

The pyramid's perpendicular height cannot be shown on a net.

So when I've unfolded this rectangular-based pyramid, I can't label the five centimetres that we were given.

So that's something to be careful of.

The perpendicular heights of the triangles do need to be known in order to get their areas.

Because it's a rectangular-based pyramid, there are two pairs of congruent triangles.

So we've got this one, which I've got a pink perpendicular height and you can see where that is on the 3D solid.

And I've also got this blue perpendicular height, which again I've labelled.

So we need those perpendicular heights of the triangular faces in order to get their areas.

So we can use half times base times perpendicular height.

So Pythagoras' theorem can be used.

So because these are perpendicular heights we can make use of Pythagoras.

So we focus on what I've labelled as A, which is the perpendicular height of the triangle, the base of six.

I'm using a right angle triangle with the perpendicular height of the pyramid and half of the width of the rectangle.

So that's what the two is.

Pythagoras' theorem means that I can find the A squared is equal to two squared plus five squared, and therefore A is square root 29, root 29.

I'm gonna do the same to find the other perpendicular heights for the other triangle.

This time I'm using the perpendicular height of the pyramid.

That's five and I'm using half of the length of the rectangle.

Because this is a right pyramid, we know that that apex is above the centre so we are halfway, it's the midpoint, which is why we can split it into two three centimetre parts B squared equals three squared plus five squared, which is B squared equals to 34.

And so B is root 34.

So we've now got the perpendicular heights of the triangles.

I can mark them on a net, but I can't mark the perpendicular heights of the pyramid.

Now we can go through and work out the surface area of the pyramid.

So A is the rectangular base.

So four times six gives us 24.

B is a triangle where we've got a base of six and a perpendicular height of root 29.

Remember we just calculated that using Pythagoras' theorem, and C is a triangle where we've got a base of four centimetres and a perpendicular height of root 34.

The surface area though is not just adding A plus B plus C, it's a plus 2B plus 2C.

We've got pairs of congruent faces.

We can simplify our thirds there and get 24 plus six root 29 plus four root 34, which is 79.

6 square centimetres to three significant figures.

So we have to do a little bit more work.

There's a little bit more involved here in order to get the perpendicular heights of those faces so we can get the surface area.

Here is a check for you.

The perpendicular height of this rectangular-based pyramid is 10 centimetres.

Calculate the perpendicular height H of the triangle with base edge eight centimetres to one decimal place.

Pause the video.

You might want to go back over the last example that I went through to work out which part you are doing, and when you're ready to check your answer, press play.

So this is 10.

We were told that the perpendicular height of the pyramid is 10.

H is this distance here from the apex to the base with the edge of eight centimetres.

And this will be three because it is a right pyramid, which means that we are on the midpoint so three.

Using Pythagoras' theorem with that right angle triangle we get H as 10.

44.

So the altitude, which is the perpendicular height of the triangle is 10.

4 to one decimal place.

Well done if you manage to get through that.

Continuing with the check, work out the surface area of this rectangular-based pyramid to one decimal place.

So you can see the altitude that you just calculated, which is 10.

44, and the altitude of the other triangular face is 10.

77.

So what is the surface area of this rectangular-based pyramid? Pause the video and then when you're ready to check your answer, press play.

So surface area is finding the total of the areas.

You need the area of the rectangle, which is the base.

You need two lots of the area of triangle one and two lots of the area of triangle two.

There are two different triangles in this pyramid.

So the area of the rectangle is easy to do eight times six.

The area of triangle one is half times eight times the perpendicular height, which is 10.

44, but you want two of them so you're doubling it.

And the area of the triangle two is six times 10.

77 times half doubled and this comes out to 196.

1 square centimetres to one decimal place.

So we're now gonna look at some questions where we actually have the surface area and what else we can calculate from it.

So a right square-based pyramid has a surface area of 144 square centimetres.

The area of one of the triangular faces is 20 square centimetres.

What other dimensions of the square? So Alex says the four triangular faces are congruent as this is a square-based right pyramid.

So because it's square based, all the bases of the triangles are the same.

And because it's a right pyramid and the apex is therefore above the centre of the square, they will be four congruent triangles.

If X is the edge length of the square, then the surface area would be x squared.

'cause that would be the area of the square plus four times 20 because we know that the area of one of the faces is 20 and we know that all four of the triangular faces are congruent.

So we can set up an equation and say that x squared plus 80 equals 144, four times 20 is where that 80 came from.

Taken away 80 from both sides and square rooting tells us that the X is eight.

X was out edge length.

So the square is eight centimetres long and eight centimetres wide because it is a square.

Here is a diagram of the square-based pyramid, and Alex says if the edge of the square is eight centimetres and the triangular face has an area of 20 square centimetres, then the perpendicular height of the triangle is five centimetres.

So where did Alex get that from? Well, the area of the triangle is half times base times perpendicular height.

We know the base is eight, so half times the eight is four.

If our perpendicular height, I've called HT and we know it equals to 20.

We know the area is 20.

So we can solve that to get the HT, which is the perpendicular height of my triangle is five.

Alex is continuing to say that if using Pythagoras' theorem he can work out that the perpendicular height of the pyramid is three centimetres.

So the perpendicular height of the pyramid would be the square root of five squared minus four squared.

Where's the four come from? Well if we can see that right angle triangle marked with black dash lines, because it's a right pyramid, the apex is above the centre.

So halfway in would be four centimetres because our square is eight by eight.

Solving that we get three.

So here is a check for you.

A square-based pyramid has a surface area of 203 square millimetres.

The square face has an area of 49 square millimetres.

What is the area of each triangular face? So pause the video and when you're ready to check, press play.

So let X be the area of each triangular face.

Then 49, the area of the square plus four x because it was a right square-based pyramid.

Then all of those triangular phases are congruent equals our area.

Solving that we get X to be 38.

5.

So each triangular face has an area of 38.

5 square millimetres.

So we're now onto the last task of the lesson.

So question one, there's two parts to it.

So part A, complete the dimensions of the net from the solid and part B using the net or otherwise calculate the surface area of the pyramid.

So pause the video and when you press play, we'll move on to the next question.

Question two part A.

Use Pythagoras' theorem to find the length marked X on this square based pyramid.

And part B is calculate the surface area of the pyramid.

Again, press pause and when you're ready to move on, press play.

Question three, calculate the surface area of this rectangular-based pyramid.

Pause the video and then when you're ready for the last question and task B, press play.

Last question then.

This square-based pyramid has a surface area of 2,352 square metres, and each triangular face has an area of 444 square metres.

What is the perimeter of the square face? Pause the video.

And when you press play, we're gonna go through the answers to task B.

Here's question one.

Part A was to complete the dimensions of the net from the solid.

So remember that the perpendicular height of the pyramid will not appear on the net.

We could put the 40 on because we can see that from the solid.

And we can put the 29, which is the perpendicular height of the triangular face.

We had to use Pythagoras' theorem to work out the edge of the isosceles triangle.

And you can do that using the 29 and a 20 because of the altitude meeting the midpoint of the base.

So 20 square plus 29 squared, and then square rooted.

For part B using the net or otherwise calculate the surface area of the pyramid, well you needed to do the area of the square base and four lots of the area of the triangle because it was a square-based right pyramid.

So 40 squared plus four times a half times 40 times 29.

And so your surface area is 3,920 square centimetres.

Moving on to question two, part A was used by Pythagoras' theorem to find the length marked X on this square-based pyramid.

That length marked X was the perpendicular height of the triangular face.

We needed to use seven as a short edge of our right angle triangle.

And again, because the apex is above the centre, so x squared, which is our hypotony use of this right angle triangle is equal to seven squared plus 24 squared.

Evaluating that to find the X is 25, so the perpendicular height of that triangle is 25 centimetres.

Part B was calculate the surface area of the pyramid.

So we need the area of the square, which we can see as 14 squared and four lots of the area of the triangle because there are four congruent triangles here, four times a half times base times height.

So 14 is the base and the perpendicular height we calculated in part A and the surface area is 896 square centimetres.

Onto three, it continues onto the next page.

There is quite involved this question.

So you needed to calculate the surface area of the rectangular-base pyramid.

So firstly, we needed to calculate the perpendicular heights of the triangular faces.

There were two pairs of congruent triangular faces because it was a rectangle.

So we had two different bases.

So I've worked out the perpendicular height of what I'm calling triangle A and that is 13.

And then the same for triangle B, which is 12 root two.

So now we've got our perpendicular heights, we're using Pythagoras' theorem and using the perpendicular height of the pyramid, we can now calculate the surface area.

So the area of the rectangle is 24 times 10.

The area of triangle one is going to be half times base times 13 or 24 times 13 and double it 'cause there are two of those.

And triangle two, there are two triangles where we are going to do half times base, which is 10 times perpendicular height, which is 12 root two, and this simplifies to 552 plus 120 root two in an exact form, or 721.

7 square centimetres.

Last question, this square-based pyramid has a surface area of 2,352 square metres and each triangular face had an area of 444 square metres.

So what is the perimeter of the square face? So thinking about how the surface area value would've been calculated, it would be the area of the square plus four lots of the area of the triangle.

So 2,352 is equal to x squared.

If we let X be the edge of the the square plus four lots of 444.

This means the x squared is 576.

So the area of the square face is 576.

We can square root that to get the edge of the square is 24.

So the perimeter would be four lots of 24, which is 96 metres.

Really well done on that task.

So to summarise today's lesson on the surface area of a pyramid, well firstly, pyramids are a group of 3D shapes.

They're the polygonal base and triangular faces that meets at an apex.

Right pyramids are where the apex is directly above the centre of the polygonal base.

Oblique pyramids are where the apex is not located directly above the centre of the polygonal base.

The surface area of the pyramid is calculated by totaling the areas of the triangular faces and the base.

And if the surface area is known, then other lengths or heights can be calculated.

Really well done on that lesson today, and I look forward to working with you again in the future.