Question

If a cube with all sides being 8 cm is dilated by a scale factor of 1/4, what would be the surface area of the dilated cube?

____ cm^2

Answer 1

To find the surface area of a cube after it has been dilated, we can follow these steps:

1. **Calculate the surface area of the original cube**:
The formula for the surface area (\(A\)) of a cube with side length \(s\) is:
\[
A = 6s^2
\]
For the original cube with a side length of 8 cm:
\[
A = 6 \times (8 \, \text{cm})^2 = 6 \times 64 \, \text{cm}^2 = 384 \, \text{cm}^2
\]

2. **Determine the new side length after dilation**:
When a shape is dilated by a scale factor \(k\), the new side length \(s'\) can be calculated by:
\[
s' = k \times s
\]
Here, the scale factor \(k = \frac{1}{4}\) and the original side length \(s = 8 \, \text{cm}\):
\[
s' = \frac{1}{4} \times 8 \, \text{cm} = 2 \, \text{cm}
\]

3. **Calculate the surface area of the dilated cube**:
Now we can use the new side length to calculate the surface area of the dilated cube:
\[
A' = 6 \times (s')^2 = 6 \times (2 \, \text{cm})^2 = 6 \times 4 \, \text{cm}^2 = 24 \, \text{cm}^2
\]

Thus, the surface area of the dilated cube is:
\[
\boxed{24} \, \text{cm}^2
\]