Surface area of a cuboid

A cuboid is a three-dimensional solid similar to a cube, whose faces are rectangles or, in special cases, squares. Here is what a cuboid looks like:

To work out the surface area of a cuboid, we find the area of all six faces and add them up. Opposite faces always have the same area. If the edges of the cuboid have lengths a, b, c, as shown in the diagram, the surface area of the cuboid is

$$S=2ab+2ac+2bc$$

which we can simplify to the formula

$$S=2(ab+ac+bc)$$

Calculator: work out the surface area of a cuboid

How did we get this?

We need to add up the areas of all the faces. Let’s start with these two opposite faces:

These faces are the rectangles ABGH and CDEF. We use the formula for the area of a rectangle and multiply the lengths of its sides. The area of rectangle ABGH is

$$S_{ABGH}=|AB|\cdot |BG|$$

We work out the area of the other rectangle, CDEF, as

$$S_{CDEF}=|CD|\cdot |DF|$$

But notice that side AB has the same length as side CD, and side BG has the same length as side DF. So both rectangles have the same area, and the two highlighted faces of the cuboid have a total area of

$$S_1=2\cdot |AB|\cdot |BG|$$

Next we work out the area of the two side faces:

We know that both faces have the same area. We find the area of rectangle BGFD as

$$S_{BGFD}=|BG|\cdot |GF|$$

and the area of both faces together is

$$S_2=2\cdot |BG|\cdot |GF|$$

That leaves the front and back faces:

The area of rectangle ABCD is

$$S_{ABCD}=|AB|\cdot |BD|$$

and the area of the whole highlighted pair of faces is

$$S_3=2\cdot |AB|\cdot |BD|$$

We find the surface area of the whole cuboid by adding up all the partial areas:

$$S=S_1+S_2+S_3$$