Lesson video

Hello, I am Mr. Gratton and thank you so much for joining me in this lesson on "Similarity and Enlargement." Today, we will look at enlargements of three-dimensional objects and how the volumes of those objects and their images are related.

Pause here to check out some important keywords for today.

The first part of this lesson will compare the volumes of cuboids before and after an enlargement.

But before we can compare volumes of 3D shapes, let's have a quick look at 2D shapes.

A linear scale factor is the number we multiply a length by on an object to find the corresponding length on the image.

So if the linear scale factor is k, then any length on an image will be k times larger than the corresponding length on the object.

Any enlarged image should be similar to its object.

Oh, and also, this is a very important thing to remember.

If the linear scale factor is k, then the area scale factor is k-squared.

The area scale factor is how many times bigger the area of an image is compared to its object.

So object A has an area of 10 centimetres squared, so the image has an area of 10 multiplied by k-squared.

2D shapes aren't really that special.

A similar relationship exists between a 3D object and its enlarged image.

The volume of a cuboid is width times height times depth, and this cuboid has a volume of 840 millimetres cubed.

This enlarged cuboid with a linear scale factor of three from the object above it, well, let's find the lengths of its sides and its volume.

Remember, every length will be multiplied by the linear scale factor.

So the height will be 8 times 3, its width, 7 times 3, and its depth, 15 times 3.

We won't evaluate these lengths yet as keeping them in this form will be pretty helpful.

Okay, the volume of the enlargement is the product of these three edge lengths, the height, the width, and the depth of that enlarged image.

But what are these three numbers? Well, the 8, 7, and 15, they're the dimensions on the original object, check on the top left to see.

And so 8 multiplied by 7 multiplied by 15 is the of that original object.

And so these three 3s belong to the linear scale factor where the linear scale factor of three is applied once each to each of the three edges of the enlarged image.

3 multiplied by 3 multiplied by 3 is how much bigger the enlarged cuboid is when compared to the original cuboid.

This simplifies to 840 multiplied by 3-cubed.

3-cubed is the volume scale factor.

A volume scale factor describes how much bigger the volume of an enlargement is when compared to the original object.

Okay, but what's so special about 3-cubed? Well, amazing Jacob has learnt to spot patterns with these scale factors.

If the area scale factor is the square of the linear scale factor, then the volume scale factor is the cube of that linear scale factor.

Amazing job, Jacob.

For this enlargement, we cubed the linear scale factor of 3 to get the volume scale factor of 3-cubed.

The volume scale factor is always the cube of the linear scale factor, no matter the value of that linear scale factor.

So for now, let's call any linear scale factor p.

Notice that all of the edge lengths on the image are the exact same edge lengths of the object just multiplied by p, that linear scale factor.

If the volume of the object is abc centimetres cubed, then the volume of the image is this, which we can rearrange to a times b times c, multiplied by p times p times p, which is simplified to abc times by p-cubed showing that no matter what the number p is, p-cubed is always the value of the volume scale factor.

Right, the linear scale factor here is 2.

Pause here to find the length, X inches.

5 inches on the object multiplied by the linear scale factor of 2 gives 10 inches on the image.

So we know that the linear scale factor is 2.

Pause here to write down the volume scale factor.

The cube of 2 is 8, so the volume scale factor is 8.

The volume of the object is 30 inches-cubed.

Pause here to use the volume scale factor to find the volume of the image.

The volume scale factor is 8.

So 8 multiplied by the volume of the object 30 gives 240 inches-cubed.

This is the volume of the image.

Here's a completely different pair of similar cuboids.

The linear scale factor is 4.

Pause here, to find the volume of that image.

Don't let the 12-foot and 60-foot lengths trick you.

They are not corresponding edge lengths, so they're actually pretty useless for this question.

If the linear scale factor is 4, then the volume scale factor is 4-cubed equals 64.

The volume of the image is therefore 100 multiplied by 64 or 6,400 cubic feet.

Keeping track of all of these edge lengths and scale factors can be pretty messy.

Putting all of this information into a scale factor table is good practise to help you keep things neat.

This is especially helpful when dealing with non-integer lengths and scale factors.

Here we have two similar cuboids.

Pause here to think about or discuss what information can you put into this scale factor table? 12 and 18 are corresponding edge lengths so they can go on the same row of this scale factor table.

So the linear scale factor is therefore 18 over 12, which simplifies to 3 over 2.

Straight away, if you know the linear scale factor, write down the volume scale factor, which is the cube of the linear scale factor.

3-cubed over 2-cubed is 27 over 8.

We also know the volume of 30, that's for shape C, so the volume of D is 30 multiplied by the volume scale factor, which when simplified is 101.

25.

Okay, this is a little different.

Object E is enlarged to make image F, but image F is smaller than E.

Well, that's because the linear scale factor is between zero and one.

If the linear scale factor is k, pause here to find the value of k.

32 over 48 simplifies to 2/3.

I think that makes sense.

2/3 is less than one, and so a scale factor of 2/3 will make the image smaller than its object.

Right, now that you've got the linear scale factor, pause here to find the volume scale factor.

2-cubed over 3-cubed is 8 over 27.

The volume of E is 7,200 centimetres cubed.

You've got all the information you need to fill in this scale factor table.

Pause now to do this and find the volume of image F.

Here's the scale factor table and 7,200 multiplied by 8 over 27 equals 2,133 after some rounding.

Nice one everyone, good effort so far.

For these practise questions, question one asks you to find the length of X, the volume scale factor, and the volume of B, and it says that the linear scale factor is a multiply by 5.

For question two, read Izzy's statement and explain why she is incorrect.

Also find the true volume scale factor from computer drawing to the printed one.

Pause here for questions one and two.

For question three, pause here to complete the scale factor table and find the volume of D.

And for question four, what do those hash marks mean on E and what does that mean for F? Pause here to calculate the volume of F and it may help you to draw a scale factor table.

Great stuff.

For the answers to question one, we have that X centimetres is 15 centimetres.

The volume scale factor is 5-cubed or 125.

And the volume of B is 20 multiplied by the volume scale factor of 125 giving 2,500 centimetres cubed.

For question two, Izzy is talking about the edge lengths being six times larger, but edges are lengths, not volumes.

So a multiplied by 6 is a linear scale factor, not a volume scale factor.

The volume will increase by a lot more than just 6.

The volume scale factor is 6-cubed or 216, so the printed cuboid is 216 times bigger.

For question three, pause here to compare scale factor tables, the volume of D is 10,290 centimetres cubed.

And for question four, pause here to compare scale factor tables, the volume of F is 145,800 centimetres cubed.

So we've seen that if a linear scale factor is p, then the volume scale factor is p-cubed.

This is true for cubes and cuboids, but is this also true for other 3D objects? Well, let's find out.

And Jacob asks the same question, how do we find the volume scale factor for other 3D shapes like these two cylinders? Well, Sam's right, let's find each of their volumes first and figure it out for ourselves.

But before that, pause here to think about or discuss, what do you think the volume scale factor is gonna be between these two cylinders? Right, from cylinder A to B, the linear scale factor is 3.

So let's multiply each length on the object by 3 to get the corresponding lengths on the image.

The height is 20 times 3 centimetres, whilst the radius on that circular face is 4 times 3 centimetres.

The volume of any 3D shape is the product of three lengths that are all perpendicular to each other.

But what does that mean for a cylinder? Well, here's the volume of a cylinder.

The three lengths that are perpendicular to each other are the radius from a circular face and another radius perpendicular to the first one.

We also have the height of the cylinder that will be perpendicular to either of the two other radii.

Pi, however is a constant, not a variable length like a radius or a height.

Constants are not affected by linear scale factors, and because they are not lengths, they do not count towards the three perpendicular lengths that make up the volume of a three-dimensional object.

Okay, let's find the volume of each cylinder in order to find the volume scale factor from A to B.

Let's substitute the radius and height of each cylinder into that formula at the top to get the volume of A at 320 pi.

And the volume of B is this.

Let's do some rearranging of this pretty messy expression.

4 times 3, all squared can be rewritten as 4 times 3 multiplied by 4 times 3.

Here are the three 3s that show the linear scale factor.

So let's rearrange this product to keep the three 3s together.

The 4 times 4 times 20 pi is the volume of A and the 3-cubed is the multiplier from volume A to volume B.

Yet again, the volume scale factor is the cube of that linear scale factor.

And Jacob is absolutely amazed.

Even though we're dealing with cylinders not cuboids, the relationship is still the same.

The volume scale factor is the cube of the linear scale factor and Sam seems to think this relationship makes sense for all three-dimensional shapes because as we discussed earlier, the volume of all 3D shapes are just the product of three perpendicular lengths multiplied by some constants where each of these three lengths when enlarged, will be affected by the linear scale factor three times, once for each length.

And so the volume scale factor is just the linear scale factor three times each time multiplied together.

These are the three perpendicular lengths of a cylinder.

For a cube, the perpendicular lengths are its width, height, and depth that are all equal in length, and it's even true for incredibly weird three-dimensional shapes like an ellipsoid whose three perpendicular lengths are its maximum width, height, and depth.

And Jacob remembers correctly because pi is a constant, not a length, it does not count towards the three perpendicular lengths.

Here's a different three-dimensional object, a sphere.

Pause here to find the linear scale factor from C to D.

36 divided by 9 is 4.

Now that we have the linear scale factor, find the volume scale factor.

Pause now to write it down.

4-cubed is 64.

And the volume of C is 972 pi.

Pause here, to find the volume of D.

972 pi multiplied by the volume scale factor of 64 gives 62,208 pi centimetres cubed.

We can find the volume scale factor even if no other scale factor is known.

All we need is one pair of corresponding lengths on an object and its image.

For example, on these two similar triangular prisms, both vertical heights are known.

The linear scale factor is therefore 14 over 10 or 7 over 5.

And so we can find out the volume scale factor by cubing the linear scale factor.

7-cubed over 5-cubed or 343 over 125.

This works for all three-dimensional shapes including real world objects and scale models of those objects.

So here's a scale model of a dragon statue and here's the real thing, the real statue that is, the linear scale factor of the tails from the scale model to its statue is 52 over 2.

6 or 20.

And so the volume scale factor is 20-cubed or 8,000.

The volume of the real statue is 8,000 times the volume of the scale model.

Here we have a very old fashioned phone and a digital model of that phone.

Some of the lengths on either phone are labelled, write down a pair of corresponding lengths on the model and real phone.

Both the 2 and 9 centimetre lengths are corresponding because they are both the antennae of each phone.

Using these two lengths, find the linear scale factor from model to real phone.

The linear scale factor is 9 over 2 or 4.

5.

The digital model has a volume of 8.

2 centimetres cubed.

Using all of this information, find the volume of the real phone.

Pause now to do this.

8.

2 multiplied by the volume scale factor of 9 over 2 all cubed is 747.

225 centimetres cubed.

Good stuff.

Here's the final few practise questions.

For question one, pause here to find the linear scale factor and volume scale factor and then find the volume of B.

And finally, question two.

Here is a scale model of a car.

The tyre on the scale model is two inches wide.

The volume of the real car must be less than the volume of 1.

3 times 10 to the power of 7 centimetres cubed, which is written in standard form.

Pause here to figure out which of these tyres are not too big for the real car.

Great effort, everyone and good job on this task.

For question one, the linear scale factor is 7 over 2, whilst the volume scale factor is 343 over 8.

The volume of A is 192 centimetres cubed, whilst the volume of B is 8,232 centimetres cubed.

And finally, question two.

The linear scale factors from the scale model tyre to each real tyre are 7.

5, 9 and 10 respectively.

Even though the tyres have been measured in inches, we can still directly apply the linear scale factor and volume scale factor to the volume of the real car even though it's written in centimetres cubed.

The volume of the real car is 13 million centimetres cubed so we can cube each of the linear scale factors found from each tyre and multiply these volume scale factors to the volume of the scale model to see if the result is greater than or less than 30 million centimetres cubed.

The scale factors from both the 15-inch and 18-inch tyres will result in a volume of less than 30 million, meaning that those two tyres could belong in the real car, but the 20-inch tyre is just too big.

And a very well done if you manage to wrap your head around that question.

And also, a very well done for your effort in all of the questions in our lesson where we have learned about the volume scale factor, a scale factor between the volumes of two similar three-dimensional shapes.

And that these 3D shapes can either be mathematical objects such as cuboids or real life objects such as a dragon statue and its scale model.

We only need one pair of corresponding lengths across two similar shapes to find the linear scale factor and volume scale factor.

To summarise, for any linear scale factor k, the volume scale factor is always k-cubed.

Once again, I thank you all for your effort today.

I have been Mr. Gratton and you have been amazing.

Until next time, take care and have an amazing rest of your day.