Express the Surface Area of a Cuboid in Terms of a: Algebraic Formula Problem

To find the surface area of a cuboid, I need to identify its three dimensions from the diagram and apply the surface area formula.

From the SVG diagram labels, the three dimensions of the cuboid are:

• Length: $a^2$
• Width: $a + 2$
• Height: $a$

The surface area formula for a cuboid with dimensions $l$, $w$, and $h$ is:
$SA = 2(lw + lh + wh)$

Let me calculate each face area:

• Face 1 (length × width): $a^2 \cdot (a+2) = a^3 + 2a^2$
• Face 2 (length × height): $a^2 \cdot a = a^3$
• Face 3 (width × height): $(a+2) \cdot a = a^2 + 2a$

Now, applying the surface area formula:
$SA = 2[(a^3 + 2a^2) + a^3 + (a^2 + 2a)]$
$SA = 2[2a^3 + 3a^2 + 2a]$
$SA = 4a^3 + 6a^2 + 4a$

Therefore, the surface area of the cuboid expressed in terms of $a$ is $4a^3 + 6a^2 + 4a$.