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Problem of the Week
Problem C and Solution
I Want More Cubes


Rashid has a wooden cube with a side length of \(10\) cm. He makes three cuts parallel to the faces of the cube in order to create \(8\) identical smaller cubes, as shown.

A horizontal cut divides the original cube into a top half and a bottom half. Two perpendicular vertical cuts then divide each half into four identical cubes.

What is the difference between the surface area of the original cube and the total surface area of the \(8\) smaller cubes?


Solution 1

Each face on the original cube has an area of \(10 \times 10=100~\text{cm}^2\). Since there are \(6\) faces on a cube, the surface area of the original cube is \(100 \times 6=600~\text{cm}^2\).

Each of the smaller cubes has a side length of \(5\) cm. So the surface area of each smaller cube is \(5 \times 5 \times 6 = 150~\text{cm}^2\). There are \(8\) smaller cubes, so the total surface area of the smaller cubes is \(8 \times 150=1200~\text{cm}^2\).

Therefore, the difference in surface area is \(1200-600=600~\text{cm}^2\).

Solution 2

Each cut increases the surface area by two \(10~\text{cm}\times 10~\text{cm}\) squares, or \(2\times 10 \times 10=200~\text{cm}^2\).

Since there are three cuts, the increase in surface area is \(3\times 200~\text{cm}^2=600~\text{cm}^2\).