Video: Finding the Diagonal Length of a Square given Its Area

Given that the area of each square on the chessboard is 81 cm², find the diagonal length of the chessboard.

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Video Transcript

Given that the area of each square on the chessboard is 81 centimeters squared, find the diagonal length of the chessboard.

So we know that the area of each of one of these small squares on the chessboard is 81 centimeters squared. And what we want to find is the diagonal length of the chessboard. Now, to enable us to do this, there are a couple of methods. So let’s have a look at both of them. Well, first of all, let’s take a look at one of the small squares. Well, if we know that the area of the small square is equal to 81 centimeters squared, we also know that the formula for the area of a square is s squared, so the side length squared.

And therefore, what we need to do is square root the area of one of these small squares to find out what the side length is. So therefore, the side length is gonna be equal to the square root of 81. Well, this is gonna be equal to nine because we don’t need to include the negative value because it’s a distance. So therefore, we can say the side length of each of the small squares is nine centimeters.

Okay, great. But how is this useful? Well, what we can do now is we can calculate the diagonal length of one of the small squares. And we can do that because if we take a look at half of our square, then this is going to be a right triangle. And for a right triangle, we can work out the length of the hypotenuse using the Pythagorean theorem. And this tells us that 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared, where 𝑐 is the hypotenuse and 𝑎 and 𝑏 are the lengths for the two shorter sides.

Well, for our calculation, what I’m gonna call is our diagonal 𝑥. And therefore, we can say that 𝑥 squared is gonna be equal to nine squared plus nine squared because each of the sides is the same length. So 𝑥 squared is gonna be equal to 81 plus 81. So therefore, we’re gonna get 𝑥 squared is equal to 162. So now, all we need to do is take the square root. And when we do that, we’re gonna get a diagonal length or 𝑥 is equal to nine root two. Well, that’s the length of one of the small squares. But what we’re looking for is the length of the total chessboard. And If we look down the diagonal length of the chessboard, we can see that we’ve in fact got eight diagonals, so eight small diagonals of the small squares.

So therefore, we can say that the diagonal length of the whole chessboard is gonna be equal to eight multiplied by the diagonal length of one of the small squares, so eight multiplied by nine root two. So this is gonna give a final answer for the length of 72 root two centimeters. Okay, great. So this was using one method. But let’s say that there was another method we could use. So let’s take a look at this now.

So for the alternate method, and again we’re wanting to find the diagonal length of the whole chessboard. What we can take a look at this time is how many squares are on the chessboard. And we can see that the chessboard is eight by eight. So eight multiplied by eight is equal to 64. So therefore, we can say that the chessboard contains 64 squares. So therefore, if we wanted to work out the total area of the chessboard, what we can do is multiply 64 by 81. And that’s because we know that the area of each square on the chessboard is 81 centimeter squared. So this would give us a total area of 5184 centimeters squared.

So therefore, if we wanted to find the side length of the chessboard, what we’d do is take the square root of 5184, which is equal to 72. So therefore, if we know that each side of the main chessboard is 72 centimeters long, then what we can do is split it up like we did with the small square and form a right triangle. And then we can apply the Pythagorean theorem. This time, I’m gonna use 𝑑 to represent our diagonal. So therefore, we could say that 𝑑 squared is gonna be equal to 72 squared plus 72 squared.

Well, it is worth noting we already know what 72 squared is because we in fact square rooted 5184 to get 72. So therefore, 72 squared must be 5184. So therefore, 𝑑 squared is gonna be to 5184 plus 5184, which gives us the final value for 𝑑 squared of 10368. So then we can take the square root of both sides of the equation. So when we do that, we’re gonna get 72 root two. So therefore, the diagonal length is 72 root two centimeters, which is the same as we got with our first method.

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