Cubes on a Cube
Weekly Problem 26 - 2016
A cube has each of its faces covered by one face of an identical cube, making a solid as shown. What is the surface area of the solid?
A cube has each of its faces covered by one face of an identical cube, making a solid as shown. What is the surface area of the solid?
Age
11 to 14
Challenge level
Problem
Image
What, in $\text{cm}^2$, is the surface area of the solid?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
The shape consists of $7$ cubes, and has a total volume of $875\text{cm}^3$. Each cube therefore has volume $875\text{cm}^3 \div 7 = 125\text{cm}^3$.
Therefore, the side length of the cube is $\sqrt[3]{125\text{cm}^3} = 5\text{cm}$, so each face has area $(5\text{cm})^2 = 25\text{cm}^2$.
Each of the outer cubes has five of its faces showing, so there are $6 \times 5 = 30$ faces showing altogether. These have a total area of $30 \times 25\text{cm}^2 = 750\text{cm}^2$.