Surface Area and Diagonal of a Cuboid

Calculates the surface area of a cuboid as 2 × (depth×width + width×height + height×depth): three pairs of matching faces. The space diagonal is also shown.

Explanation

A cuboid has six faces, and opposite faces are identical rectangles. So you need the areas of only three different faces, each counted twice.

S = 2(ab + bc + ca)

$S$ — the surface area
$a$ — the depth
$b$ — the width
$c$ — the height

The line joining the two most distant corners, the space diagonal, comes from applying the Pythagorean theorem twice.

d = \sqrt{a^2 + b^2 + c^2}

Example

With the defaults, $a = 4$ , $b = 5$ and $c = 6$ :

the three face areas are $ab = 20$ , $bc = 30$ and $ca = 24$
surface area $S = 2 \times (20 + 30 + 24) = 2 \times 74 = 148$
space diagonal $d = \sqrt{16 + 25 + 36} = \sqrt{77} \approx 8.7750$

Watch out

Surface area is a sum of face areas, so its unit is a length squared (cm²), not a length cubed like the volume (cm³). Forgetting the factor of 2 and stopping at $ab + bc + ca$ halves the answer. The diagonal reported here is the space diagonal, the one running through the inside of the solid, not a diagonal drawn on a single face. All three edges must be positive.