Surface Area of Cuboid

Theorem

Let $\CC$ be a cuboid whose edges are of length $a$, $b$ and $c$.

The surface area $S$ of $\CC$ is given as:

$S = 2 \paren {a b + b c + a c}$


Proof

Each of the faces of $\CC$ are rectangles:

$2$ of these faces are adjacent to edges of length $a$ and $b$
$2$ of these faces are adjacent to edges of length $b$ and $c$
$2$ of these faces are adjacent to edges of length $a$ and $c$.


Hence from Area of Rectangle:

there are $2$ faces with area $a b$
there are $2$ faces with area $a c$
there are $2$ faces with area $b c$.

Hence the result.

$\blacksquare$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Rectangular Parallelepiped of Length $a$, Height $l$, Width $c$: $4.27$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): parallelepiped
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): parallelepiped
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Rectangular Parallelepiped of Length $a$, Height $b$, Width $c$: $7.27.$
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