# Stretched surfaces

The edges of a cube are stretched, can you find the new surface area?

## Problem

The height, width and length of a cube are multiplied by 2, 3 and 6 respectively to create a cuboid.

The surface area of the cuboid is $N$ times the surface area of the original cube.

What is the value of $N$?

This problem is taken from the UKMT Mathematical Challenges.

## Student Solutions

Image

The original cube has $6 $ faces, so its surface area is $6$ original faces.

Using a diagram can help show how the cube grows:

Image

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The cuboid's faces are $2$ by $3$ original faces, $2$ by $6$ original faces and $3$ by $6$ original faces, as shown in the diagram, and each appears twice.

The surface area of the cuboid is $2(2 \times 6 +2 \times 3 +3 \times 6)=72$ original faces.

The surface area of the cuboid is $N$ times the surface area of the original cube.

$72=6\times12$

so $N=12$

so $N=12$