The diagram shows a cuboid with labelled side lengths. Nick says, “The volume of any solid made with four of these cuboids is 1080 centimetres cubed.” Part (a) asks us, “Is Nick right?” You must justify your answer.
So a solid is made with four of these cuboids. All we guaranteed that whatever solid is made, its volume will be 1080 centimetres cubed. That’s what we want to know: is Nick right? Well, the volume of the solid made from four of these cuboids is four times the volume of a single cuboid. It doesn’t matter how we arrange these four cuboids. As no extra material is added and none is removed, the total volume will be the same.
So now, the question is “what is the volume of one of our cuboids?” Well, we’re given the side lengths of the cuboid. They are nine centimetres, five centimetres, and six centimetres. Now, the volume of the cuboid is given by its length times its width times its height. And so the volume of our cuboid is nine times five times six. And as those lengths are all in centimetres, this volume is in centimetres cubed.
And we can enter four times nine times five times six into our calculators. And we find that this value is 1080. So the volume of any solid made with four of these cuboids is 1080 centimetres cubed, as Nick said. So to answer the question “Is Nick right?” yes, Nick is right.
We can move on to part (b).
Part b (i) Draw another cuboid which could be made with four of these cuboids. Label the dimensions of the new cuboid on the diagram.
So we have four of these nine-centimetre-by-five-centimetre-by-six-centimetre cuboids and we want to use them to form a new cuboid. Here’s a diagram of four of our cuboids lined up in a row. Together, they form a longer cuboid.
We are asked to label the dimensions of our new cuboid. You’ll note that I’ve already marked the width and the height, but we need to find the length as well. In this arrangement, the length of our new cuboid is nine centimetres plus nine centimetres plus nine centimetres plus nine centimetres.
The original cuboids are lined up. And so we just add their lengths to find the length of the new cuboid. And this sum is nine centimetres times four which is 36 centimetres. Let me emphasize that this is just one way of arranging four of the original cuboids to form a new cuboid. We could have stacked them some different way, getting a different diagram with different dimensions.
There’s more than one correct answer to this part of the question. But if we stick with our original choice, all we have left to do is to erase some edges of our original cuboid, which are not edges of this new cuboid. Doing so makes it clear that this is a cuboid in its own right. So this is our answer to part bi. As we said before, there are many other equally correct answers to this part of the question.
The final part of our question is part b (ii): calculate the surface area of your new cuboid.
To do this, it helps to draw in the hidden faces of our cuboid, like so. The surface area of our cuboid is the sum of the areas of the rectangular faces of our cuboid. We see that two of the rectangular faces have a width of five centimetres and a height of six centimetres. Each of them has area five times six centimetres squared. And so together, they contribute two times that.
The top and bottom faces have a length of 36 centimetres and width of five centimetres. Together, they contribute two times 36 times five centimetres squared to the surface area of the cuboid. And the remaining two faces are the front and back faces with length of 36 centimetres and height six centimetres. Each of them has an area of 36 times six centimetres squared. And so together, they contribute two times that to the surface area.
We can put each of these into our calculator. Two times five times six is 60, two times 36 times five is 360, and two times 36 times six is 432. Adding his up again using a calculator, we get a surface area of 852 centimetres squared. And this is our answer to part b ii.
One thing to note here is that while this is the surface area of the new cuboid we chose to draw in part b i, it isn’t the surface area of all possible cuboids we could have drawn in part b i. If you drew a different cuboid in part b i and didn’t get 852 centimetres squared in part b ii, you might still be right. The surface area of your cuboid might just be different from the surface area of mine. Surface area is different from volume in this way.
We showed in part a of the question that the volume of any solid made with four of these cuboids is 1080 centimetres cubed. However, it isn’t true that the surface area of any solid made with four of these cuboids will be 852 centimetres squared.