The total surface area of a cube is 384 centimetres squared. Calculate the volume of the cube.
It’s important to be really clear on the definitions for both surface area and volume. The surface area of a three-dimensional shape is the total area of all of its faces. The units for surface area will always be squared, for example, centimetres squared, metres squared, and so on.
The volume is the amount of space the shape can hold and its units will always be cubed, centimetres cubed, metres cubed, and so on. The shape we’re interested in is a cube. Remember a cube has six identical faces, each of which is the shape of a square. The sum total of the area of these six faces is 384 centimetres squared. So we can work out the area of one of the faces by dividing 384 by six.
Let’s use the bus stop method to help us. We can’t divide three by six in this method. So instead, we ask ourselves how many sixes we need to make 38. That’s six sixes with a remainder of two since six sixes are 36. Next, we ask ourselves how many sixes make 24. That’s four. So 384 divided by six is 64.
We now know then that the area of each of the faces of our cube is 64 centimetres squared. Now, remember we said that each of the faces was a square and now we know that each square has an area of 64 centimetres squared.
Since the area of a square can be found by squaring its width, we can do the opposite to that to find the width of each of the squares. We find the square root of 64, which is eight. We now know then that all the dimensions of our cube are eight centimetres. Now, we have all the measurements we need to find its volume.
The formula for the volume of a cuboid is width times length times height. Since a cube is simply a cuboid with all of its dimensions the same, we can write this as the volume of a cube is equal to its width cubed. The volume of our cube then is eight cubed. That’s eight times eight times eight. Since eight times eight is 64, the volume of our cube can be found by multiplying 64 by eight.
We can use any formal written method for multiplication. Let’s use the column method. Four multiplied by eight is 32. So we put the units part of this calculation in the first column from the right. We carry the three. Six multiplied by eight is 48. And when we add the three, we get 51. That means then that 64 multiplied by eight is 512.
The volume of our cube then is 512 centimetres cubed.