Video: Finding the Area of a Square Using Diagonals

In this video, we will learn how to find the area of a square in terms of its diagonal length.

14:21

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.

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