Lesson video

Hello everyone.

welcome and thank you for joining me, Mr. Gratin in this lesson on enlargement and similarity.

Today we will look at the relationship between the volume scale factor, the linear scale factor, volumes and lengths between a pair of three dimensional similar shapes.

Pause here to have a look at some of the keywords that we'll be using today.

First up, let's find lengths when given the volumes of two similar 3D shapes.

Let's say we had a multiplier of K between corresponding edges on two similar 3D shapes like four and 4K on these two similar cuboids or 1.

5 and 1.

5k on these two prisms or 70 and 70k on these two three dimensional robots, then the multiplier between the volumes of each pair of similar 3D shapes will be K cubed.

So for example, if the volume of this smaller prism is 10 cubic inches, then the volume of this larger prism is 10k cubed with the units still being cubic inches.

The multiplier between lengths is called the linear scale factor, whilst the multiplier between the volumes of two 3D shapes is the volume scale factor.

So if the linear scale factor between two shapes is four, pause now to find the volume scale factor.

The volume scale factor is the cube of the linear scale factor.

So four cubed is 64 and for these two similar shapes, pause here to find both the linear scale factor and the volume scale factor.

The pair of corresponding edges are the 10 centimetre and 30 centimetre edges.

Therefore, the linear scale factor is three, because 10 multiplied by three is 30 and the volume scale factor is three cubed or 27.

Calculating lengths, volumes and scale factors can be effective on a scale factor table.

Here are two similar pyramids.

Let's see if we can place corresponding lengths and volumes on this scale factor table.

The lengths or vertical heights of 20 centimetres and four centimetres are corresponding to each other.

However, the length of 28 centimetres is corresponding to neither, so it does not belong on the same row as these two corresponding values.

The linear scale factor is 20 over four, which simplifies to a multiply by five.

As soon as you know the linear scale factor qubit to find the volume scale factor five cubed is 125.

We also know the volume of C, and therefore we can use the volume of C and the volume scale factor and multiply them together to get the volume of D at 6,000 centimetres cubed.

Here we have two cylinders, they are similar to each other.

Pause here to complete the scale factor table and identify the volume of cylinder F.

The volume of cylinder F is 87,122 centimetres cubed.

We can find the linear scale factor if we are given the volumes of two similar shapes.

The linear scale factor is the cube root of the volume scale factor.

For example, the volumes of 1040 and 130 will give a volume scale factor of eight, and if the volume scale factor is eight, then the linear scale factor is the cube root of eight and the cube root of eight is two.

We have found the linear scale factor from G to H, even though no lengths on either the object or the image are known.

For this check the volume of shape K is 1,331 times the volume of shape J.

Pause here to calculate the volume and linear scale factors from J to K.

The volume scale factor is 1,331, whilst the linear scale factor is the cube root of 1,331 at 11.

And pause here to find the values of A to D in this scale factor table.

Here are the answers.

The volume scale factor is 512, meaning that the linear scale factor is the cube root of 512 at eight.

But what's the point in finding a linear scale factor if we're not going to use it, well, we can use the linear scale factor to find lengths on either one of a pair of similar shapes, even if the linear scale factor is found from volumes.

There's a lot of information here between these two similar shapes.

Remember, my best advice is to look out for corresponding pieces of information.

The volumes of 1,920 and 30 are corresponding to each other because they're both volumes of similar shapes and the lengths of 20 inches and X inches are also corresponding lengths to each other.

We can put corresponding values into the same row of a scale factor table, the corresponding volumes in a volume row, giving a volume scale factor of 1,920 over 30, which simplifies to 64.

The moment you know a volume scale factor, cube root it in order to find the linear scale factor.

The linear scale factor is therefore four.

And so we can use the linear scale factor of four alongside the width of 20 on the image and divide 20 by four to get a width of five on the object, which therefore means that X equals five.

Remember, the same linear scale factor applies to all lengths, so the heights will also have a linear scale factor of four.

So if the height on the object is 2.

5, the height on the image will therefore be 10.

For this check, use this scale factor table to help you identify the lengths of the edges P and Q.

The volume scale factor is 729 and it's cube root, the linear scale factor is nine.

P centimetres and 45 centimetres are a pair of corresponding lengths whilst two centimetres and Q centimetres are also a pair of corresponding lengths.

Therefore, P centimetres equals five centimetres whilst Q centimetres equals 18 centimetres.

Great effort so far everyone for question one of this practise task.

Find both the linear scale factor and volume scale factor from A to B and find the volume of object A, onto question two as well.

Write down a calculation that shows that the volume scale factor is 421.

875 and use the volume scale factor to then find the linear scale actor, pause now for these two questions.

For question three, pause here to complete this scale actor table and find the lengths of the sides R centimetres and P centimetres.

And finally question four, find the lengths of the edges labelled X, Y and Z.

It may help you to construct a scale factor table.

Pause now for question four.

Nice work, great perseverance on some pretty tricky questions.

For question one, the answers are the linear scale factor is eight and the volume scale factor is 512.

The volume of A is 3,200 divided by the volume scale factor giving 6.

25 centimetres cubed.

For question two, the volume scale factor is found from a division of 101,250 and 240.

The linear scale factor is the cube root of this at 7.

5 or 15 over two.

For question three P centimetres equals 50 centimetres whilst R centimetres equals 8.

5 centimetres.

Pause here to compare this completed scale factor table to your own.

And finally, question four X centimetres equals 52 centimetres Y centimetres equals five centimetres, whilst Z centimetres equals 114.

4 centimetres.

Pause here to compare this complete scale factor table to your own calculations.

So we've calculated the edge lengths of similar 3D shapes from their volumes and the volume scale factor, but why? Let's look at how we can use volumes and edge lengths to find the surface area of an object.

But what is a surface area? Well, it is the sum of the areas of each face on a 3D shape for a polyhedron, it's the sum of the areas of each of its faces where each face is a polygon.

Here's an example, a cuboid, it has six faces and the top and bottom faces are a congruent pair, so are the front and back faces and there's also a third different congruent pair, they are the left and right faces.

12 times 20 is the area of the top face.

We can multiply 12 times 20 by two to account for the congruent bottom face that we cannot see on this diagram.

12 times five is the area of the front face and multiplying this by two will also account for the congruent back face.

Same here, 20 times five times two accounts for both the areas of the left and right faces.

The surface area of the cuboid is therefore the sum of all three of these calculations which represent all six faces of this cuboid.

Each calculation can get evaluated and then added together to get 800.

The surface area of the entire cuboid is 800 centimetres squared.

Remember, because we're dealing with an area, the units are centimetres squared, not cubed like a volume.

Here we have a different cuboid with three faces visible on this diagram.

Pause here, to calculate the area of each Of the three rectangular faces, the top face is 11 multiplied by 30.

The front face is 30 multiplied by 12 and the right face is 11 multiplied by 12.

Now we know the areas of the three faces, pause here to find the surface area of the entire cuboid.

The other three faces that we cannot see are congruent to the top front and right faces.

So we can add together 330, 360 and 132 to find the total area of the three visible faces and then double the answer to account for the three faces that we cannot see.

Alternatively, we could double 330, double 360 and double 132 individually and then add together the three results.

Either way, we should get a total surface area of 1,644 centimetres squared.

We can use the linear scale factor to find lengths on an object or image and then use those lengths to find the area of a face.

And if we do not know the linear scale factor, we might be able to figure it out from the volume scale factor.

Like with these two triangular prisms, we know the height of A and the width of B, so we cannot find the linear scale factor, because we do not know a pair of corresponding lengths.

However, we do know both volumes, so we can place both of these volumes into the scale factor table to find the volume scale factor.

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

If the vertical height on A is 14, then the vertical height X on B is 42 and if the width on B is nine, then the width on A, Y is three.

Now we have the width and vertical height on both triangular faces, we can find the areas of each triangular face.

The area of the triangular face on A is 14 multiplied by three, then divided by two equals 21 centimetres squared.

Pause here to find the area of the triangular face on B.

The area of the triangular face on B is 42 multiplied by nine, divided by two, which equals 189 centimetres squared.

All the known information about these two cuboids have been placed into this scale factor table by filling in appropriate parts of the scale factor table, find the area of the top face E on cuboid C.

Pause now to do this.

The first most important thing to find is the volume scale factor of 125.

Therefore the linear scale factor is five.

The top face needs a width and a depth, so we need to divide the width on D by five to get the corresponding width on C, that's 30 centimetres.

We already know the depth on C at 48 centimetres and so the area of face E is 30 multiplied by 48 or 1,440 centimetres squared.

When in doubt, fill in the whole scale factor table and figure out what you need after it is all filled in and similar again, pause here to find the area of the triangular face on G.

The triangular face requires a vertical height and a width, so let's fill out all those parts on the scale factor table and the volume part of the scale factor table.

The area of the triangular face on G is 378 centimetres squared.

We can find the surface area of two similar shapes if we can find the linear scale factor and use enough lengths that are known across both shapes.

Now in order to find surface areas, a lot of information needs to be dealt with probably now more than ever, keeping that information tidy on something like a scale factor table is ever so helpful.

Let's see if finding the surface area of J is even possible with the given information, we know both volumes and so the volume scale factor is 27 over eight.

This also means all linear scale factors are known at 1.

5 or three over two.

Okay, let's start looking at some lengths.

We know the vertical height on K, and so we divide 7.

5 by the linear scale factor to get five on J.

This is the same for the width on K giving us a width on J of 12.

Again, the hypotonus or right diagonal on the triangular face K is known, which then when divided by 1.

5 gives us 13 on J.

The depth on J is already known at 24 centimetres, but to complete the scale factor table we can multiply 24 by 1.

5 to get 36 the depth on K.

Okay, let's label J with all of the lengths that we need to calculate the surface area, the vertical height of five, the width of 12 and the right diagonal of 13.

Some of you might have spotted that the sides of the triangular face make a Pythagorean triple well done if you did, we should have enough lengths to find the surface area, but let's just check for certain.

The five faces of this prism are the two congruent triangles and three different rectangles.

The top rectangle is 24 multiplied by 13.

The bottom rectangle, even though we cannot see it, we know it's two perpendicular lengths, are 24 and 12 and similar for the left face at 24 multiplied by five.

Notice how each of these three rectangles have the depth of 24 as part of its calculation and the area of one of the triangular faces is five multiplied by 12 divided by two.

But because there are two triangular faces, we then double the answer to account for both of the faces or more simply, you could just do five multiplied by 12 without the divide by two and multiplied by two.

Either way, the two triangular faces have a total area of 60 centimetres squared.

The surface area is then the sum of all of these areas at 780 centimetres squared.

Okay, let's find the surface area of cuboid L by first finding the edge lengths on both cuboids.

Pause here to fill in the scale factor table.

Here are the answers.

Pause here to compare your answers to the ones on screen.

And now using these edge lengths, calculate the areas of the three visible faces on L, the top front and right faces, pause here to do so.

And the areas are on the top.

We have 70 centimetre squared on the front, 147 centimetre squared and on the right 210 centimetre squared.

Nearly there using these three areas.

Pause here to find the surface area of L.

We take these three areas and add them together.

Then double the sum to account for the other three faces we cannot see to get a total surface area of 854 centimetres squared.

Great stuff, good application of all of the information that you know for the final few practise questions.

Pause here for question one, which asks you to fill out a scale factor table and use appropriate lengths to find the surface area of cuboid B.

And similarly for question two, but you have to be a bit more careful about which lengths help you find the areas of which faces.

Pause now to fill in this scale factor table and find the surface area of C.

And lastly, question three, pause here to find the surface area of F and feel free to construct your own scale factor table to help you.

Bravo everyone.

Great, perseverance on some really elaborate questions.

For question one the surface area of B is 4,752 square yards.

Pause here to check all of your calculations and compare them to the ones on screen.

For question two pause again here to check all of your calculations on this scale factor table.

Leading to a surface area on C of 510 square millimetres.

And for question three, a very well done if you spotted that the hypotonus on the triangular face F was 102.

5 centimetres long via Pythagoras theorem, the linear scale factor was 2.

5 from E to F.

The total surface area of F was 49,500 square centimetres.

Pause here to check all of your information and compare it to everything on screen.

Unbelievable efforts everyone on this question and all of the other questions in a really challenging lesson where we have found a linear scale factor K by cube routing, the volume scale factor K cubed, where the volume scale factor can be found by identifying the multiplicative relationship between volumes of two similar 3D shapes and where we can use volumes of similar shapes to find their surface areas.

By first of all, calculating edge lengths.

Once again, thank you all so much for your effort today.

I have been Mr. Grattan and you have been spectacular.

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