Emptied Cube
Weekly Problem 26 - 2015
What are the volume and surface area of this 'Cubo Vazado' or 'Emptied Cube'?
What are the volume and surface area of this 'Cubo Vazado' or 'Emptied Cube'?
The sculpture 'Cubo Vazado' [Emptied Cube] by the Brazilian artist Franz Weissmann is formed by removing cubical blocks from a solid cube to leave the symmetrical shape shown.
Image
![Emptied Cube Emptied Cube](/sites/default/files/styles/large/public/thumbnails/content-id-10174-15_weekly26.png?itok=U6M1mVS7)
If all the edges have length $1$, $2$ or $3$ units, what is the volume of the sculpture and what is its surface area?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Volume of the sculpture
Consider the sculpture to consist of three layers, each of height $1$ unit. Then the volumes of the bottom, middle and top layers are $5$ cube units, $2$ cube units, and $5$ cube units respectively. So the volume of the sculpture is $12$ cube units.
Alternatively: the sculpture consists of a $3\times 3\times 3$ cube from which two $2\times 2\times 2$ cubes have been removed. The $2\times 2\times 2$ cubes have exactly one $1\times 1\times 1$ cube in common - the cube at the centre of the $3\times 3\times 3$ cube. So the volume of the sculpture is
$$27\;\mathrm{units}^3-\left(2\times 8\;\mathrm{units}^3- 1\;\mathrm{units}^3\right) =12\;\mathrm{units}^3\,.$$
Image
![Emptied Cube Emptied Cube](/sites/default/files/styles/large/public/thumbnails/content-id-10174-15_weekly26_sol.png?itok=uGHHXH35)
Surface area of the sculpture
In each of six possible directions, the view of the sculpture is as sown,
with the outer square having side $3$ units and the inner square having
side $1$ unit. So the surface area of the sculpture is
$$6\times 8\;\mathrm{units}^2 = 48\;\mathrm{units}^2\;.$$