What is the surface area of a similar cuboid with a scale factor of 3?

The surface area of a similar cuboid with a scale factor of 3 is 9 times the original.

When dealing with similar shapes, the scale factor affects different dimensions in specific ways. For a cuboid, which has three dimensions (length, width, and height), if the scale factor is 3, each dimension of the new cuboid is 3 times the corresponding dimension of the original cuboid. This means that if the original cuboid has dimensions \( l \), \( w \), and \( h \), the new cuboid will have dimensions \( 3l \), \( 3w \), and \( 3h \).

The surface area of a cuboid is calculated using the formula \( 2lw + 2lh + 2wh \). For the original cuboid, this would be \( 2lw + 2lh + 2wh \). For the new cuboid, with dimensions \( 3l \), \( 3w \), and \( 3h \), the surface area formula becomes \( 2(3l \cdot 3w) + 2(3l \cdot 3h) + 2(3w \cdot 3h) \).

Simplifying this, we get:
\[ 2(9lw) + 2(9lh) + 2(9wh) = 18lw + 18lh + 18wh \]

Notice that \( 18lw + 18lh + 18wh \) is exactly 9 times the original surface area \( 2lw + 2lh + 2wh \). Therefore, when the dimensions of a cuboid are scaled by a factor of 3, the surface area increases by a factor of \( 3^2 = 9 \). This principle applies to any similar shapes: the surface area scales by the square of the linear scale factor.