Question Video: Finding the Area of a Square Mathematics • 6th Grade

Find the difference between the area of a square whose side length is 17 cm and the area of a square whose diagonal length is 20 cm.

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Video Transcript

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.

So in this problem, what we have are two squares, and what we need to do is work out the area of each of these squares so we can compare them and find the difference between their areas. So first of all, to find the area of a square, we know it’s equal to 𝑠 squared, so the side length squared. So what we’re gonna do is label our squares A and B. So first of all, what we can do is take a look at square A. And we know the side length of square A. So therefore, we could say that the area of square A is gonna be equal to 17 squared, which is 289 centimeters squared.

Okay, great! So that’s area A. So now, if we take a look at our second square, well what we can do is split this into two parts because we’ve got a diagonal and both of these are gonna be right triangles. So if we call our side length 𝑠, then what we could do is work out our side length, so therefore, work out our area. And to do this, what we’re gonna do is use the Pythagorean theorem. And what the Pythagorean theorem states is that 𝑐 squared equals 𝑎 squared plus 𝑏 squared, where 𝑐 is the hypotenuse, so the length of the longest side, and 𝑎 and 𝑏 are the lengths of the shorter sides.

So therefore, if we take a look at square B, we’re gonna have 20 squared equals 𝑠 squared plus 𝑠 squared. And that’s because as it’s a square, both of the shorter sides are the same length. So therefore, we can say that 400 is equal to two 𝑠 squared. So now, what we could do is divide both sides of the equation by two. So when we do that, we get 200 is equal to 𝑠 squared. Well now, what we could do is, in fact, take the square root to find out the length of a side. However, we don’t need to do this. And that’s because we’ve got 𝑠 squared. And if we take a look at our formula, we know that the area of the square is gonna be equal to 𝑠 squared. So therefore, what we can say is the area of square B is gonna be equal to 200 centimeters squared.

Okay, great! So we now have the area of square A and the area of square B, but have we solved the problem? Well, no because what we’re looking to find is the difference between them. So let’s find this out now. Well, the difference between them is going to be equal to 289 minus 200 because that’s the area of square A minus the area of square B. Well, this is gonna give us 89. So therefore, we can say that 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 is 89 centimeters squared.