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Geometry Difficulty 3.7 AMC 10/12 Find the answer Canada

A large $5\times 5\times 5$ cube is formed using 125 small $1\times 1\times 1$ cubes. There are three central columns, each passing through the small cube at the very centre of the large cube: one from top to bottom, one from front to back, and one from left to right. All of the small cubes that make up these three columns are removed. What is the surface area of the resulting solid?

$204$
$206$
$200$
$196$
$192$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The original $5 \times 5 \times 5$ cube has 6 faces, each of which is $5 \times 5$ .

When the three central columns of cubes is removed, one of the “ $1\times 1$ squares” on each face is removed.

This means that the surface area of each face is reduced by 1 to $5 \times 5 - 1 = 24$ . This means that the total exterior surface area of the cube is $6 \times 24 = 144$ .

When each of the central columns is removed, it creates a “tube” that is 5 unit cubes long. Each of these tubes is $5 \times 1 \times 1$ .

Since the centre cube of the original $5 \times 5 \times 5$ cube is removed when each of the three central columns is removed, this means that each of the three $5 \times 1 \times 1$ tubes is split into two $2 \times 1 \times 1$ tubes.

The interior surface area of each of these tubes consists of four faces, each of which is $2 \times 1$ . (We could instead think about the exterior surface area of a $2 \times 1 \times 1$ rectangular prism, ignoring its square ends.) Thus, the interior surface area from 6 tubes each with 4 faces measuring $2 \times 1$ gives a total area of $6 \times 4 \times 2 \times 1 = 48$ .

In total, the surface area of the resulting solid is $144 + 48 = 192$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.