The surface of a cube

A cube is a three-dimensional solid similar to a cube whose faces are rectangles or squares. Let's see what a cube looks like:

To calculate the surface area of a cuboid, we need to calculate the area of all six faces of the cuboid and add them together. The area of the opposite faces are always the same. If the cube has side lengths equal to a, b, c, as shown in the figure, then the surface of the cube is equal to

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

which can be further simplified to the formula

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

Calculator: calculate the surface area of the cube

How did we figure this out?

We need to add up the area of all the walls. First we can take these two opposite walls:

These walls are formed by the rectangles ABGH and CDEF. So we use the formula to calculate the area of a rectangle by multiplying the lengths of the sides of the rectangle. So we find the area of the rectangle ABGH by multiplying the lengths of the rectangle by the lengths of the

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

We calculate the area of the second rectangle CDEF as

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

But we can notice that the length of the side AB is the same as the lengths of the side CD and the length of the side BG is the same as the length of the side DF. So the area of both rectangles will be the same. It is therefore true that the area of the two highlighted sides of the cuboid will be equal to

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

Next, we calculate the area of the two side walls:

We know that the area of both walls will be equal. So we calculate the area of the rectangle BGFD as

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

The area of both walls will then be equal to

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

It remains to calculate the area of the front and back walls:

The area of the rectangle ABCD will then be equal to

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

and the area of the entire highlighted area will be equal to

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

We calculate the surface area of the whole cube by adding all the partial contents:

$$\Large S=S_1+S_2+S_3$$