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 use our understanding of similarity as well as the linear, area, and volume scale factors in order to solve complex problems and problems involving percentages and ratios.

Pause here to have a quick look at some important keywords.

First up, let's look at problems that require the linear, area, and volume scale factors.

It is possible to find the linear scale factor, the area scale factor, and the volume scale factor even if we only know one pair of corresponding values on a 3D object and its similar image.

These corresponding values could be lengths, areas, or volumes, including edge lengths and surface areas.

For example, we know the areas of corresponding faces on these two prisms. From this pair of areas, we can find the area scale factor, and therefore we can find both the linear scale factor and the volume scale factor.

Let's have a look at this on a scale factor table.

We are given two areas, so let's place these corresponding values into the areas row of the table.

130 and 6,370 give the area scale factor 49.

The square root of an area scale factor is the linear scale factor.

Going from an area scale factor to a linear scale factor is a much more straightforward process than going from an area scale factor straight to a volume scale factor.

The square root of 49 is 7, and then the cube of the linear scale factor is the volume scale factor; therefore, 7 cubed is 343.

So from just one pair of corresponding faces, we have found the volume scale factor of 343.

Okay, here we have two cylinders.

They are similar to each other.

Pause here to take the two pieces of given information and place them into the correct parts of this scale factor table.

These are both areas, specifically the areas of a circular face.

Because they are corresponding areas, their values go into the same areas row.

Now, using these two areas, pause here to calculate the area scale factor.

5,840 divided by 365 equals 16.

The area scale factor is 16.

And finally, pause here to calculate the linear scale factor and volume scale factor.

My advice is to always find the linear scale factor first.

This can be done by finding the square root of the area scale factor, and the square root of 16 is 4.

Once you have found the linear scale factor, you can then cube it to find the volume scale factor at 4 cubed or 64.

We can use the three scale factors to find lengths, areas, and volumes on one of the two similar shapes, but only if the corresponding value on the other similar shape is already known or can be found out.

For example, we have the volumes of these two shapes, but only one area, the area of shape E.

The corresponding area on shape F is the area of that face that is currently unknown.

Let's find the area of that unknown face.

We have two volumes giving us a volume scale factor of 27/8.

My advice still holds.

Always aim to find the linear scale factor first.

We can cube root the volume scale factor to find the linear scale factor.

And the cube root is 3/2 or 1.

5.

We can then square the linear scale factor to get the area scale factor of 9/4.

We know an area on E and so the corresponding area on F is the area of E multiplied by the area scale factor, giving 234.

The area of that face on F is 234 centimetres squared.

Here we have two similar metal nuts.

By using all of the given information about their areas and volumes, pause here to fill in as much of the scale factor table as you can.

The two volumes give a volume scale factor of 125/27, leading to a linear scale factor of 5/3 and an area scale factor of 25/9.

The area on G of 18 leads to an area on F of 50 millimetres squared.

Jacob's confusion is really understandable.

We have so many different values to try and understand, so which words and contexts describe a length, area, or volume? Well, words that describe a measurement can help show what dimension it is.

For example, perimeters, heights, and distances all describe lengths that are related to a linear scale factor.

Whereas areas can be areas of either a single face or the surface area of an entire object.

No matter the type of area given, it will always be related to an area scale factor.

And volumes can be described as a capacity, the amount of space inside of an object.

Furthermore, sometimes the context can help but you have to be careful about what each context means.

For example, the distance travelled implies a length, whilst a shape, house, or object being covered in paint, ink, or wallpaper, that sort of thing usually implies that the paint is covering an area.

The space taken up by an object is found from its volume, or, for example, the amount of water that the object displaces also calculates the volume of that object.

And lastly, the index of the unit can sometimes tell you the dimension of that measurement.

Centimetres, inches, and kilometres all measure distances, whilst centimetres squared, with an index of 2, describes a two-dimensional value, the area.

Similarly, centimetres cubed, with an index of 3, describes a 3D value on a 3D shape, the volume.

So for example, here we have two rugby balls.

The size of each ball is determined by the distance in inches from one end on the surface of the ball to the other end.

Jacob notices that this description involves both a distance and an area since it references a surface.

So is the size of the ball determined by a length or its area? Pause here to think about or discuss how the size of the ball is measured.

Whilst it does mention a surface, it is actually talking about a distance across the surface.

So the size of the ball is determined by a length across the curved exterior of that rugby ball.

We can confirm this by looking at the units.

These units describe a length in inches, not an area in inches squared.

So the sizes of 9 inches and 16 inches are both lengths.

These two rugby balls will now be covered in ink for logos and sponsors.

The 9-inch ball will be covered in 126 millilitres of ink.

So let's find out how much ink the larger ball will need.

And Jacob is absolutely correct.

This ink will cover a surface.

So now, we're dealing with areas, and finding the area scale factor is absolutely essential.

Millilitres is a measurement of capacity for liquids or fluids.

There's no such thing as millilitres squared or cubed, so you cannot use the index on this unit to identify its dimension.

However, this context talks about how the ink will cover an area or cover a surface and so its value is still linked to areas.

Right, let's take all of this information and place it into the correct part of this scale factor table.

Do not get caught in a trap.

Both 16 and 9 are square numbers and so it is easy to think that these numbers are instantly linked to square units or areas.

Well done if you did spot that these are square numbers.

And sometimes square numbers can give you a hint that their purpose is linked to square units or areas.

However, do not automatically assume that they are.

It is still important to assess from the context and other information whether these values are linked to areas or whether they are linked to lengths or volumes.

In this context, both 9 and 16 are in inches, which are lengths, so we place them in the lengths row of our scale factor table.

The linear scale factor is therefore 16/9, giving an area scale factor of 256/81.

We know the amount of ink needed to cover the area of the smaller rugby ball, so multiplying this value by the scale factor gives approximately 398, where 398 millilitres is the amount of ink required to cover the surface of the larger rugby ball.

Okay, here we have two balls dropped into a bowl of water.

Both balls displace the water.

Pause here to figure out which dimension of measure is described by this scenario.

Displacement describes volume, the amount of space a shape takes up.

The bigger the object, the more water it displaces.

Lovely stuff.

Onto some practise questions.

For question one, pause here to fill in the scale factor table and find the volume of B.

And for question two, pause here to construct your own scale factor table to find a length, a surface area, and a volume.

And finally, question three, this is a very challenging question that requires you to pay a lot of attention to the context given.

Millilitres and litres are used in two very different ways here.

Pause now to find the maximum capacity of a fuel tank.

Superb effort, everyone.

The answers for question one.

The volume of B is 1,024,000 centimetres cubed.

Pause here to check your calculations on this ratio table.

For question two, X centimetres equals 88 centimetres.

The surface area of D is 3,630 centimetres squared and the volume of C is 280 centimetres cubed.

For question three, the heat-proof coating is applied to the exterior surface of the fuel tank.

Therefore, the heat-proof coating is linked to areas whilst the fuel goes inside the tank, describing a capacity or volume of the fuel tank.

The maximum capacity of car E's fuel tank is 40 litres.

Pause here to compare your calculations to these onscreen.

Rather than being given, for example, two corresponding volumes, what if we are given corresponding values described as percentages of each other or through the use of a ratio? Well, let's find out.

Relationships between two similar objects can be written as percentages or a ratio.

Representing this information on a scale factor table can help identify the appropriate scale factor.

Here we have two bowling balls with surface areas in a ratio 9:25.

9 and 25 are not their actual surface areas, rather a description of how much bigger the surface area of one ball is compared to the other.

On this scale factor table, we can place the surface area ratio in the area row.

You will notice that with these ratios that the scale factor between them is just a fraction with one part of the ratio over the other part of the ratio, but maybe with some simplifying involved as well.

In this case, the scale factor from 9 to 25 is the fraction 25/9, therefore the linear scale factor is 5/3 and the volume scale factor is 125/27.

The volume of the duck pin bowling ball is 324 centimetres cubed.

This is an actual measurement, an actual volume, not just part of a relationship like the ratio was.

The volume of the 10 pin bowling ball is therefore 324 multiplied by the volume scale factor at 1,500 centimetres cubed.

Okay, for this check, here we have two eggs.

The amount of egg in the medium versus large eggs are in the ratio 1,000:1,331.

First of all, figure out whether this ratio describes a length, area, or volume and pause here to complete as much of the scale factor table as possible.

The ratio describes a volume and so the volume scale factor is 1,331/1000.

The linear scale factor is therefore 11/10, meaning that the area scale factor is 121/100.

If the surface area of the large egg is 68, then the surface area of the medium egg is 68 divided by the scale factor at 56.

2 centimetres squared.

We know nothing about the lengths of the eggs, however the linear scale factor was still helpful in finding the area scale factor.

Two shapes that are similar aren't always written as being similar, but you can potentially identify their similarity from the types of shapes that they are.

For example, all circles are similar, all squares are similar.

If two hexagons are regular, then they are also similar.

This is the same for any two regular pentagons or regular decagons.

Any two regular polygons with the same number of sides are always similar.

This is also true for 3D shapes as well.

Any two cubes are similar, and there are other examples, such as any two spheres being similar to each other.

Some measurements can be described as a percentage of a corresponding measurement on a similar shape.

It is always important to reference one measurement as 100%, with the other measurement being greater than or less than 100%.

For example, the area of square B is 81% of the area of square A.

Pause here to think about or discuss which square is smaller and how do you know.

On this scale factor table, square B is 81%, meaning that square A is the original 100%.

This means that square B is smaller than square A because its percentage size is less than 100%.

Percent just means divide by 100.

So when calculating a scale factor, these two divide by one 100s cancel each other out.

The area scale factor is 81/100, not 81/100%.

The linear scale factor is therefore 9/10, with the 10 being the A part of the ratio and the 9 being the B part of the ratio.

This is consistent with the area values where the area percentage of A was bigger and so the ratio part of A should also be bigger even if it's talking about the lengths like the perimeter.

The ratio is 10:9, not 9:10.

You might have noticed that 10 and 9 are just the square roots of 100 and 81, the number parts of each percentage.

100 and 81 are proportions for an area so we can square root them straight away to find the proportions for a length like a perimeter.

The square roots of 100 and 81 are 10 and 9, giving the exact same ratio of 10:9 as before without needing to explicitly calculate any scale factors.

Whilst calculating scale factors isn't always necessary when going from a proportion of areas to a proportion of lengths, sometimes it is still helpful due to the percentages given not being easy to square root or cube root.

We also still need to calculate scale factors if we need to calculate an actual length, or an actual area, or an actual volume.

For example, here we have two prisms. The volume of prism D is 337.

5% the volume of C.

D has a percentage of 337.

5, so C has a percentage of 100.

The volume scale factor is 337.

5/100, which simplifies to 27/8.

Feel free to check this on a calculator.

Therefore, the linear and area scale factors can be found.

The ratio of surface areas is just the ratio using that scale factor at 4:9.

4 is the C part of the ratio because C had the smaller volume percentage at 100%, not 337.

5%.

Furthermore, we are given the surface area of C at 160 square inches, so we still need to use the scale factor of 9/4 in order to find the surface area of D at 360 square inches.

For this check, we have two circles.

Pause here to figure out which two parts of this scale factor table the values 100% and 125% should go.

F has the percentage of 125%, so E is the original 100%.

The ratio 100:125 can be simplified.

Pause here to fully simplify this ratio.

25 is a factor of both and so the fully simplified ratio is 4:5.

As they are both proportions, it is possible to simplify them on a scale factor table in a similar way to a ratio.

Pause here to find the values of A and B on the area row.

Area proportions are the square of the length proportions, therefore the ratio of areas is 16:25.

Now, we are given an actual measurement, the area of E at 176 centimetres squared.

Pause here to find the area of F.

Since we're dealing with areas, we can find the area scale factor by first finding the linear scale factor.

176 multiplied by the area scale factor gives 275 centimetres squared.

We can also look at similar shapes where one is a percentage increase or decrease over the other.

For example, H is 125% bigger than G.

G is the object with the original area of 100% whilst H is the image because it is 125% larger.

So if G is 100%, then H is 125% larger at a total of 225%.

As we are dealing with proportions, we can simplify 100 and 225 like a ratio, giving 4:9.

The square root of the area proportions gives the length proportions and the cube of the length proportions give volume proportions.

Therefore, the ratio of volumes is 8:27.

Here we have a red vase that has a volume that is 48.

8% smaller than the blue vase.

Pause here to think about what two percentages represent these two vases and where would you put these values in this scale factor table.

We are dealing with volumes.

The blue vase is the original, with a volume of 100%, and the red vase has a volume that is 100 takeaway 48.

8%.

We have 151.

2 as volume proportions, which can therefore be written as the ratio 100:51.

2.

Pause here to fully simplify this ratio.

I would first multiply each part of the ratio by 10, giving 1,000:512.

And then divide both parts of the ratio by 8, giving 125:64.

And finally, pause here to find the values of A, C, D, and F.

The cube roots of 125 and 64 are 5 and 4.

Squaring these proportions gives 25 and 16.

Right, that was a lot to take in.

Let's practise these proportional skills.

For question one, pause here to calculate the length of the scale model and find out how much paint is needed to cover the real car.

And now, pause here for questions two and three.

And finally, question four.

These three cuboids are similar to each other.

Pause here to find the ratio of the volumes of B and C and find the volume of C given the volume of A.

Superb effort, everyone.

For question one, the scale model is 5.

22 metres and 7.

65 litres of paint is needed.

For question two, the pentagon's could be regular.

This guarantees that they are similar.

If the area of one pentagon is 84, then the other could have been multiplied by or divided by 36.

And because the linear scale factor is 6, the larger pentagon has a perimeter of 210 centimetres.

And for question three, the ratio of perimeters is 100%:70%, which simplifies to 10:7, therefore, the ratio of areas is the square of 10 and 7 at 100 and 49.

If the area of square A is 560, then the length of one of its sides is the square root of 560.

The perimeter of square B is 66.

3 centimetres.

Pause here to compare the calculations on screen to your own.

And finally, question four.

A is the original cuboid, with a surface area of 100%, meaning that B is 49% and C is 196%.

The volumes of B and C are in the ratio of 343:2,744, which can be simplified quite neatly to 1:8.

Cuboid C has a volume of 3,430 centimetres cubed.

And pause here one last time to compare your calculations to the ones on screen.

Amazing work, everyone, on this really challenging problem solving lesson where we have found one scale factor, either the linear, area, or volume scale factor, and used that scale factor to find the other two.

And then we've used all three scale factors to find lengths, areas, and volumes on a 3D shape.

And then we went even further beyond by looking at lengths, areas, and volumes between similar shapes that have been described by ratios or percentages.

Once again, a simply amazing effort, everyone.

I have been Mr. Gratton, and you have been an incredible problem solver.

Until next time, take care and goodbye.