Video Transcript
In this video, we’ll find the area
of a square given the diagonal length. We’ll also see how we can find the
diagonal length given the area. Let’s begin with a quick recap of
what we should already know. Firstly, we know that a square is a
quadrilateral or four-sided shape with all four sides the same length. In a square, all the interior
angles will also be 90 degrees. If we’re given the side length of
the square, let’s call it 𝑙, then we know that each side would also be the side
length 𝑙.
To find the area of a square, we
would multiply the length by the length, which would be 𝑙 times 𝑙. We could also write this as 𝑙
squared. But what happens if, instead of
being given the length of the square, we’re given the diagonal length, which we
could call 𝑑? How then can we work out the
area? Let’s begin by seeing how we could
work out the side length of this square and then using this formula to help us find
another formula for the area of a square using the diagonals.
Let’s remember that a square has
got four 90-degree angles. This means we can create a right
triangle within our square and, therefore, we could apply the Pythagorean
theorem. This theorem applies to right
triangles only. And it says that the square of the
hypotenuse is equal to the sum of the squares on the other two sides.
Applying the Pythagorean theorem on
our triangle then, the hypotenuse will be 𝑑, so we’ll begin with 𝑑 squared is
equal to 𝑙 squared plus 𝑙 squared. This would of course simplify to 𝑑
squared equals two 𝑙 squared. And then remember that we’re trying
to find the length 𝑙 in terms of the diagonal. This means we’ll need to divide
both sides of the equation by two. Therefore, 𝑑 squared over two
equals 𝑙 squared. So to find 𝑙, we take the square
root of both sides, which leaves us with 𝑙 equals the square root of 𝑑 squared
over two.
Let’s remember that we’re trying to
find the area of a square which we can do now we’ve got 𝑙 in terms of the diagonal
length 𝑑. We would take our formula and fill
in the values for 𝑙 to give us that the area of the square is equal to the square
root of 𝑑 squared over two multiplied by the square root of 𝑑 squared over
two.
When we have the square root of a
number multiplied by itself, we’re just left with that number. In this case, we’ll be left with 𝑑
squared over two. We have now found a formula for the
area of a square using the diagonal length. That is, the area of a square is
equal to 𝑑 squared over two, where 𝑑 is the diagonal length. As we go through the rest of this
video, we’ll mostly be looking at squares where we’re given the diagonal length. But of course, it’s always
important to remember the formula for the area using the side length. Let’s look at our first
question.
Find the area of a square whose
diagonal is nine centimeters.
Let’s begin by modeling our
square. We know that it will have four
equal sides and four angles with 90 degrees. We might remember that it’s easy to
find the area of a square if we know the length of one of the sides. However, here we’re given the
length of the diagonal. So we’ll need to remember a
different formula, that is, that the area of a square is equal to 𝑑 squared over
two, where 𝑑 is the diagonal length.
We’re given here that the diagonal
is nine centimeters. So we can fill this into the
formula so that the area will be equal to nine squared over two. As nine squared is nine multiplied
by nine, we’ll get the answer of 81 over two. Our units here will be squared
centimeters. And so we’ve found the area of a
square whose diagonal is nine centimeters, which is 81 over two square
centimeters. Of course, we could’ve also given
our answer as the decimal 40.5 square centimeters.
So far in this video, we’ve seen
how we can find the area of a square given the diagonal length. But can we find a way to find the
diagonal length if we’re given the area? Well, we can in fact do this by
rearranging the formula for the area of a square given the diagonal. If we let the area be equal to the
letter 𝐴, then what we really need to do is rearrange it so that we have 𝑑 as the
subject of this equation. When we multiply both sides of the
equation by two, we would get two 𝐴 is equal to 𝑑 squared. In order to find 𝑑, we would take
the square root of both sides of the equation, which gives us the square root of two
𝐴 is equal to 𝑑.
We now have another formula here to
find the diagonal length 𝑑 given the area 𝐴 of a square. Let’s see how we can put that into
practice in the following question.
Given that the area of each square
on the chessboard is 81 square centimeters, find the diagonal length of the
chessboard.
Here, we’re told that the area of
each of these smaller squares on the chessboard would be 81 square centimeters. This wouldn’t just include all the
black squares, for instance, but it would also include every single white square as
well. We need to find out the diagonal
length of the chessboard. And we should recall that there’s a
formula which links the diagonal length of a square with its area. The formula is that 𝑑, the
diagonal length, is equal to the square root of two 𝐴, where 𝐴 is the area. We can get this formula by
rearranging the area of a square formula. That area is equal to 𝑑 squared
over two, where 𝑑 is the diagonal.
Before we can use the formula that
𝑑 is equal to the square root of two 𝐴, we’ll firstly need to find the area of the
entire chessboard. Rather than counting every single
square to see how many squares there are on this chessboard, we could count simply
that there’s eight squares in each row and eight squares in each column. This would also confirm that we do
indeed have a square chessboard. As we have eight squares in each
row and eight in each column, then the total number of squares will be 64
squares. To find the total area of all of
these squares, we know there’s 64 squares and each of them will have an area of 81
square centimeters. Evaluating this would give us a
total area of 5184 square centimeters.
As an alternative method to
calculate this, if we’d imagine that one square is 81 square centimeters in area,
that would mean that the length of each of these squares would be nine
centimeters. As we have eight squares along the
length of each of the side of the square and eight times nine is 72, then we could
work out the area. The area here would be the length
by the length, which would be 72 multiplied by 72, which would also give us an area
of 5184 square centimeters. We can fill in the value that the
area 𝐴 is equal to 5184 into our formula, being careful that the square root also
includes this value of the area and not just the two. We can simplify this to get 𝑑 is
equal to the square root of 10368.
Rather than reaching for our
calculator, let’s see if we can simplify this value 10368 by seeing if there’s a
square factor of it. We could write that this is equal
to the square root of 144 times 72. Breaking this in to two separate
square roots, we can evaluate the square root of 144 as 12. But, of course, 72 also has a
square factor. We can write the square root of 72
as the square root of 36 times two. We could then simplify our
calculation of 12 times six times the square root of two to 72 root two. We can then give our answer that
the diagonal length of the chessboard is 72 root two. And the units will be centimeters
as we’re working with a length rather than an area.
Let’s have a look at another
question.
Find the diagonal length of a
square whose area equals that of a rectangle having dimensions of 10 centimeters and
35 centimeters.
The best place to begin here is by
modeling our two shapes, the square and the rectangle. The rectangle has a length and a
width of 10 centimeters and 35 centimeters. And we need to find the diagonal
length of the square. The information that we’re given to
allow us to work out the diagonal length is that the area of our two quadrilaterals
is the same. We don’t have any length
information about the square, so let’s see if we can work out the area of this
rectangle.
We can recall that the area of a
rectangle is equal to the length multiplied by the width. We would then fill in our two
values of 35 and 10, and working out 35 multiplied by 10 is nice and simple, 350
square centimeters. And so the area of our square will
also be 350 square centimeters. We’ll need to remember a formula
that connects the area of a square with its diagonal. The area of a square is equal to
the diagonal 𝑑 squared over two. As we want to find the diagonal
given the area, then we can use the rearranged form of this formula to give us that
𝑑 is equal to the square root of two 𝐴, where 𝐴 is the area of the square.
We can fill in the value for the
area that we know, keeping the letter 𝑑 as that’s the unknown that we want to find
out. 𝑑 is equal to the square root of
two multiplied by 350, which simplifies to 𝑑 is equal to the square root of
700. Assuming we’re using a
non-calculator method, we’ll need to simplify this square root further. We should hopefully notice this
nice square factor of 100. And therefore, we can break down
our calculation into the square root of 100 multiplied by the square root of seven,
which simplifies to 𝑑 equals 10 root seven. As 𝑑 is the diagonal length, then
we can give our final answer of 10 root seven centimeters.
Let’s look at a final question
where we’ll need to use both of the formulas for the area of a square.
Find the difference between the
area of a square whose side length is 17 centimeters and the area of a square whose
diagonal length is 20 centimeters.
In this question, we’re told that
there’s two squares. We’re given the side length of one
and the diagonal length of the other. So let’s begin by drawing out these
squares. But don’t worry if they’re not
quite to scale. Let’s say that this first square is
the one whose side length is 17 centimeters. And of course, because it’s a
square, we know that all the sides will be 17 centimeters. And the second square, we could
make with the diagonal length of 20 centimeters.
We’re asked about the areas of
these squares. In fact, we’ll need to work out the
difference between the areas. But let’s begin by thinking how we
would find the area of each square. In the first square, we’ve got the
side length, so let’s recall our first formula. The area of a square is equal to
the length multiplied by the length or, alternatively, the length squared. We can fill in the side length 𝑙
into our formula, so we’ll be calculating 17 multiplied by 17. Using whatever multiplication
method, we’ll get the answer of 289. And because it’s an area, we’ll
have the area units of squared centimeters.
Now that we’ve found the area of
this first square, let’s see how we can find the area of the second square, where
we’re given the diagonal length instead of the side length. We can remember the second formula
for the area of a square that says that it’s equal to 𝑑 squared over two, where 𝑑
is the diagonal length. Plugging in the diagonal length 𝑑
as 20 centimeters, we have the formula that the area is equal to 20 squared over
two. 20 multiplied by 20 will give us
400 and dividing by two gives us the area is 200 square centimeters.
We mustn’t forget at this point
that we haven’t answered the question. We still need to find the
difference. We therefore take the larger area
of 289 and subtract 200, which gives us our final answer that the difference between
the areas of these two squares is 89 square centimeters.
Let’s now summarize what we’ve
learned in this video. We began by recapping what should
be a familiar formula, that the area of a square is equal to 𝑙 times 𝑙, where 𝑙
is the side length. We then saw how we could use the
Pythagorean theorem to actually develop another formula for the area of a square
given the diagonal length 𝑑, that is, that the area of a square is equal to 𝑑
squared over two. We then saw how we can rearrange
that formula 𝐴, the area, equals 𝑑 squared over two to give us the formula 𝑑
equals the square root of two 𝐴. This formula allows us to more
quickly find the diagonal length given the area of a square.
