The surface area of a large cube is $5400 \mathrm{~cm}^{2}$. This cube is cut into a number of identical smaller cubes. Each smaller cube has a volume of $216 \mathrm{~cm}^{3}$. How many smaller cubes are there?
(A) 25
(B) 125
(C) 164
(D) 180
(E) 216
Solution 1

The large cube has a total surface area of $5400 \mathrm{~cm}^{2}$ and its surface is made up of 6 identical square faces. Thus, the area of each face, in square centimetres, is $5400 \div 6=900$.

Because each face is square, the side length of each face is $\sqrt{900}=30 \mathrm{~cm}$.

Therefore, each edge of the cube has length $30 \mathrm{~cm}$ and so the large cube has a volume of $30^{3}=27000 \mathrm{~cm}^{3}$.

Because the large cube is cut into small cubes each having volume $216 \mathrm{~cm}^{3}$, then the number of small cubes equals $27000 \div 216=125$.

## Solution 2

Since the large cube has 6 square faces of equal area and the total surface area of the cube is $5400 \mathrm{~cm}^{2}$, then the surface area of each face is $5400 \div 6=900 \mathrm{~cm}^{2}$.

Since each face is square, then the side length of each square face of the cube is $\sqrt{900}=30 \mathrm{~cm}$, and so the edge length of the cube is $30 \mathrm{~cm}$.

Since each smaller cube has a volume of $216 \mathrm{~cm}^{3}$, then the side length of each smaller cube is $\sqrt[3]{216}=6 \mathrm{~cm}$.

Since the side length of the large cube is $30 \mathrm{~cm}$ and the side length of each smaller cube is $6 \mathrm{~cm}$, then $30 \div 6=5$ smaller cubes fit along each edge of the large cube.

Thus, the large cube is made up of $5^{3}=125$ smaller cubes.

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