Video Transcript
What will happen to the total volume and the total surface area of a cube being split into eight equally sized pieces, as shown in the diagram? (A) The total volume is the same and the total surface area is eight times larger. (B) The total volume doubles and the total surface area doubles. (C) The total volume doubles and the total surface area is the same. (D) The total volume is the same and the total surface area doubles. (E) The total volume is eight times larger and the total surface area is eight times larger.
We are asked to find out what happens to the total volume and total surface area when a cube is divided into eight equally sized smaller pieces. After we have calculated the total volume of the large cube and its total surface area and the total volume of the small cubes and their total surface area, we can work out a volume-to-surface- area ratio for the large cube and for the smaller cubes. This volume-to-surface-area ratio influences the reactivity of a material, and this is particularly important when looking at nanoparticles. Let’s clear some space to answer this question.
Let’s imagine this cube is two centimeters by two centimeters by two centimeters. When it is split into eight equally sized pieces, each new small piece will be one centimeter by one centimeter by one centimeter. We can measure the volume of the cube by taking the length multiplied by the width or breadth multiplied by the height. And for a cube, all these dimensions are the same. So we get two centimeters multiplied by two centimeters multiplied by two centimeters, which gives us a total volume of eight centimeters cubed. We can do the same calculation for one of the smaller cubes to get its volume. And we get one centimeter for the length multiplied by one centimeter for the width multiplied by one centimeter for the height, which gives us a volume of one of the small cubes of one centimeter cubed.
But we want to know the total volume of the eight small cubes. To do this, we must multiply the volume of a small cube by eight since there are eight cubes, and we get a total volume of eight centimeters cubed. Do you notice that the total volume has not changed? We could put the eight small cubes back together to make the original big cube of the same size and volume as before it was divided into eight. So far, we know what has happened to the total volume. It has not changed.
Now, we can calculate the total surface area of the large cube. We can work out the area of one side and multiply this value by six since there are six sides to a cube. The area of a cube is the length times the width, or we could say the length times the height or the height times the width. It does not matter, since these three dimensions are equal in size for a cube. We could even say length times length. Whichever way we do it, we will get two centimeters multiplied by two centimeters multiplied by six, which gives us 24 centimeters squared for the total surface area of the large cube, which is the area of this side and this side and this side and the three other sides that we cannot see.
In the same way, we can work out the surface area of one of the small cubes. We can say length times length times six, which is the surface area of this side plus this side plus this side plus the three sides which we cannot see. And we get one centimeter multiplied by one centimeter multiplied by six, which gives us six centimeters squared for the surface area of one small cube. However, there are eight cubes and we need the total surface area for all eight cubes. So we can multiply the surface area of one small cube by eight, and we get 48 centimeters squared, which is the total surface area of all the small cubes.
We now know what happens to the total surface area. It doubles when the large cube is split into eight smaller, equally sized cubes. Knowing this and that the total volume is the same, we can now select the correct answer to the question. Let’s put the answer options back.
What will happen to the total volume and total surface area of a cube being split into eight equally sized pieces, as shown in the diagram? The answer is (D): the total volume is the same and the total surface area doubles.
We mentioned earlier that we can take the volume and surface area and put them into a ratio and that this ratio influences the reactivity of a material. We know that the total volume for the large cube and the total volume for the small eight cubes is the same but that the surface area for the eight smaller cubes is double that of the large cube. So the volume-to-surface-area ratio is 𝑉 as to 𝑥 for the large cube and 𝑉 as to two 𝑥 for the small cubes. The smaller cubes are much more reactive than the large cube because of this ratio. And that is why nanoparticles are so reactive and so useful to us because of this large surface area and very small particles.
