Geppetto needs to paint a batch of cube shaped building blocks for his toy shop. Each cube has a side length of five centimeters. Geppetto buys a pot of blue paint which can cover an area of 2000 centimeters squared. Part a) Using this information, work out how many building blocks Geppetto can completely cover with blue paint.
When considering the amount of paint required to cover each cube, we are really thinking about the total area of all the faces of the cube. This is known as the surface area. To find the surface area, we should first think about exactly how many surfaces — that’s faces — the cube has.
We know that a cube has six identical faces. So in fact, we simply need to find the area of one of its faces and we can multiply this number by six to get the total surface area of the cube. The formula for area of a rectangle is width multiplied by height.
A square is just a special type of rectangle, in which the width and the height are the same length. So we can say that the area of the square is its width multiplied by its width or its width squared. Since the width of our square is five, the area is given by five squared.
When we square a number, we multiply it by itself. So the area of one face is 25 centimeters squared. We said earlier that there are six faces. So the total surface area of this cube can be found by multiplying 25 by six. That tells us that the surface area of the cube is 150 centimeters squared. The can of paint can cover an area of 2000 centimeters squared.
So to work out the number of cubes that can be covered by this paint, we can divide 2000 by the amount required to cover one cube; that was 150. 2000 divided by 150 is 13.3 recurring or 13 and one-third.
We actually want to know how many cubes can be completely covered. So this extra bit of a cube — the third of a cube — doesn’t count. Geppetto can only cover 13 cubes with paint.
Geppetto decides to paint a different batch of cube shaped building blocks instead. The new cubes have a side length of four centimeters instead of five centimeters. Part b) Explain how this might affect the number of blocks that he can completely cover using his pot of blue paint.
Smaller dimensions on the side lengths of the square will mean a smaller area. In fact, each face of the cube will have an area of four squared; that’s 16 centimeters squared. This means the total surface area of each cube will also be less his paint will actually go further than it did before.
We can say then that he should be able to paint more blocks with his pot of blue paint.