Video Transcript
In this video, we’re going to look
at similar figures and the relationship that exists between their areas.
So, first, let’s look at what is
meant by the term similar figures. The definition’s on the screen
here. Two figures are similar if, first
of all, they have to be the same shape, so both squares or rectangles or triangles
for example. And secondly, corresponding lengths
on these shapes have to be in the same ratio. So, if you think about a triangle
for example, the bases of these two triangles must be in the same ratio as the
heights of the two triangles.
Let’s look at an example of this
using rectangles. So, the two rectangles on the
screen here, they’re similar rectangles. And you can see this if you look at
the ratio between corresponding pairs of sides. If I look at the width of the
rectangle first of all, so the two and the three, this will give me what’s known as
a length ratio of two to three.
If I look at the other dimension of
the rectangle, so the four to six, then this would tell me that the length ratio is
four to six. But, of course, this can be
simplified by dividing both sides of this ratio by two. And if I do that, you’ll see that
it simplifies to the same ratio of two to three. So, because the corresponding pairs
of sides are both in this ratio two to three when simplified, that means that these
two rectangles are similar.
Now, in this video we want to look
specifically at the relationship between the areas of these similar figures. So, I’ll work out both of the
areas. Now, as they’re both rectangles,
that’s relatively straightforward. I just need to multiply their two
dimensions together. So, I have eight centimetres
squared for the first rectangle and 18 centimetres squared for the second.
Now, let’s use those areas to write
down an area ratio between the two rectangles. So, the area ratio is going to be
eight to 18, but that will simplify cause again I can divide both sides of the ratio
by two. So, that tells me that my
simplified area ratio is four to nine. Now, the key point is how does this
relate to the length ratio. Well, the length ratio, remember,
was two to three and the area ratio is four to nine. And perhaps you’ll spot that
there’s a relationship between those sets of numbers, and it’s a square
relationship. Two squared gives us four, and
three squared gives us nine. So, another way of writing this
area ratio would be two squared to three squared.
Now, it isn’t a coincidence that
this relationship exists. It’s illustrative of a general rule
when looking at the areas of similar figures. And the general rule is this. If the length ratio between two similar figures is 𝑎 to 𝑏, then the area ratio
between them will be 𝑎 squared to 𝑏 squared. So, this rule always holds true
when working with similar figures. And now, we’ll see how we can use
it to answer some questions.
So, here’s the first question.
We’re given a diagram of a polygon,
in fact it’s a rectangle, on a coordinate grid. And we’re asked to find the area of
a similar polygon where 𝐴 dash 𝐷 dash is equal to six.
So, let’s just look at the
rectangle that we’re given first of all. Well, the length of the base here
from 𝐴 to 𝐵, that is two units. And the length of the vertical side
here from 𝐴 to 𝐷, that is three units. So, we can fill in those two
measurements. We can also work out the area of
this rectangle, so it’s gonna be two times three. It’s gonna be six square units.
Now, let’s think about this second
polygon. So, we’re told that it’s similar,
but we’re told that the length of 𝐴 dash 𝐷 dash is six. Whereas in our current polygon, the
length of 𝐴𝐷 is three units. This means we can write down the
length ratio between these two polygons. So, using that pair of
corresponding sides, it’s going to be three to six. But of course, that ratio can be
simplified. I can divide both sides of it by
three. So, my simplified length ratio is
going to be one to two. What this means then, in practical
terms, is that all of the lengths in the enlarged polygon are twice as long as they
are in the existing polygon.
Now, we want to know about the area
of the larger polygon. So, we need to recall that general
rule that we saw before about the link between the length ratio and the area
ratio. And the rule, remember, was
this. If the length ratio is 𝑎 to 𝑏,
then the area ratio is 𝑎 squared to 𝑏 squared. So, that means I can use my known
length ratio to work out the area ratio. I just need to square both sides of
it. So, my area ratio is one squared to
two squared, which of course is just one to four.
What this means then is that the
area of this enlarged polygon is four times the area of the smaller polygon. So, I have all the information I
need in order to be able to calculate the area of this similar polygon. The area of the smaller polygon was
six square units. And if this one is four times
larger, than I just need to multiply six by four. So, the area of this polygon six by
four, which is, of course, 24 square units.
So, a quick recap of what we did
then. We used a pair of corresponding
lengths to write down a length ratio. We then used our general rule to
transform that to an area ratio by squaring both sides. Because we knew the area of the
smaller polygon, we then combined it with the area ratio to work out the area of the
larger polygon.
Okay, the next question that we’re
going to look at says, corresponding sides of two similar polygons are 18
centimetres and 25 centimetres. Given that the area of the smaller
polygon is 486 centimetres squared, we are asked to determine the area of the larger
polygon.
So, let’s think about how to
approach this then. We’re given this pair of
corresponding sides, which means we can start off by writing down the length ratio
between these two similar polygons. And so, here it is. The length ratio is 18 to 25, and
that doesn’t simplify any further. Now, we’re asked about areas, so we
need to work out the area ratio. And remember that general rule, we
need to square both sides of this in order to work out the area ratio. So, the area ratio will be 18
squared to 25 squared. And so, this is 324 to 625, which
again doesn’t simplify any further.
So, let’s use this area ratio to
work out the area of the larger polygon. So, I’m gonna give it a letter. I’m gonna call it A. Now, what this ratio means is if I
take the area of the larger polygon, which is A, and divide it by the area of the
smaller polygon, 486, then they’re in this ratio of 324 to 625. Which means I get the same result
if I were to do 625 divided by 324. So, what I’ve done here is set up
an equation which I can then solve in order to work at the area of this larger
polygon, using the area ratio that I know.
The first step to solving this
equation then is I need to multiply both sides of the equation by 486, as that’s
currently in the denominator on the left-hand side. So, I’ll have 625 over 324
multiplied by 486. And if I evaluate that and put in
the units, it tells me that the area of the larger polygon is 937.5 square
centimetres.
So, exactly the same process as in
the previous example. We wrote a length ratio first of
all, squared both sides to form an area ratio. The bit that’s different from the
last example is that previously we just had to multiply the area by four, as it was
just a one-to-four ratio. This time the ratio was a bit more
complex, so we had to transform it into a fractional relationship, use that to form
an equation, and then solve that equation to find the missing area. Okay, let’s look at a different
type of question.
We’re told that the two polygons
below are similar. And we’re asked to calculate the
value of 𝑥. Now, 𝑥 is this missing length here
in the smaller polygon.
So, let’s look at what we know. We haven’t been given a pair of
corresponding lengths this time. We’ve been given a pair of
corresponding areas. Cause we can see that the two areas
are 35 and 315 centimetres squared. So, we’re going to have to approach
this question in a slightly different way. Rather than writing down a length
ratio to start off with, we’re gonna instead right down an area ratio.
So, the ratio of these two areas is
35 to 315. That can be simplified cause both
sides can be divided by 35. So, that gives me an area ratio of
one to nine. Now, we’d like to work backwards
from knowing this area ratio to working out the length ratio. So, remember the general rule that
we saw previously. It was that whatever the length
ratio is, to get the area ratio, you have to square both sides.
Now, we’re gonna be working back
the other way from the area ratio to the length ratio. So, in order to go back the other
way, rather than squaring both sides, we need to square root both sides. So, the length ratio is the square
root of one to the square root of nine, which, of course, is just one to three. So, what this tells us is that all
the of lengths in the larger polygon are three times the lengths in the smaller
polygon.
Now, if I wanted to set it up as an
equation, just to demonstrate again what I did in the previous example. Then, I know that if I do 𝑥
divided by 18, so that corresponding pair of sides, then I’m gonna get the same
result as if I do one divided by three. So, that’s using this length ratio
of a third. So then, I can solve this equation
by multiplying both sides by 18. And so, 𝑥 is equal to six. Another way of thinking about it
without having to write down the equation is because it’s just a one-to-three ratio,
then I could just divide 18 by three to get this value of six. So, in this example, it was
slightly different. We had to start with an area ratio
and then work backwards by square rooting in order to calculate the length ratio
that we needed.
Okay, the final question says
shapes X and Y are similar with sides in the ratio five to four. If each side length is tripled,
what is the ratio of the areas of the enlarged shapes?
Now, you may have a gut feel for
what the answer should be, or you may think perhaps there’s a detail you haven’t
considered. Let’s go through some working out
in order to see how we should approach this. So, we already have a length ratio
for shapes X and Y. It’s five to four. That would mean that the area ratio
for X and Y before we triple the sides and enlarge them, well, remember, we would
square both sides of this ratio, so the area ratio for X and Y would be 25 to
16.
Now, let’s think about what happens
when we triple the side lengths. Well, we don’t know that these
lengths are five and four, just that they’re in that ratio. But if we triple them, then they
would be in the ratio 15 to 12. But here’s the key point. Because we’ve tripled both of them,
that ratio of 15 to 12 just simplifies straight back down to five to four again. So, tripling the side lengths
actually has no effect whatsoever on the length ratio because we’ve done the same
thing to both of the similar shapes.
Therefore, the area ratio of these
enlarged shapes is gonna be the same as the area ratio of the original shapes. So, the area ratio is still 25 to
16. Now, that may have been your gut
feel. But if it wasn’t, then I hope that
by going through the working out you’ll have been convinced of why this is the
case. So, the key point from this example
is if a question asks you about doubling or tripling the sides of both of the
similar shapes, then it has no impact on the length ratio and, consequently, no
impact on the area ratio. So, you can just work with the
original ratios that you’re given.
In summary then, we’ve seen the
relationship between the length ratio and the area ratio of similar figures. We’ve seen how to apply this to
calculating the area of a similar figure. And we’ve also seen how to apply it
to calculating missing lengths by working backwards from knowing the area ratio to
calculating the length ratio.
