# How to calculate the area of ​​a rectangular parallelepiped

A rectangular parallelepiped is a learned name to designate a 6-sided object, ultimately very common, that everyone calls… a box! A brick, a shoebox, a concrete block, these are examples of rectangular parallelepipeds! The surface of such an object is the sum of the external surfaces that compose it. "How much wrapping paper will I need to wrap this shoebox?" This paper surface is the surface of the parallelepiped. Presented in this way, this problem is not as complicated as it seems.

## Steps

### Part 1 of 2: find the total area #### Step 1. Locate the length, width and height

Any rectangular parallelepiped has a length, a width and a height. Make a sketch and indicate with a letter the three main sides by putting an "L" for the length, an "l" for the width and an "h" for the height.

• If you're not sure how to do this, take one of the angles and label each of the sides that converge there.
• Consider a box which has a base of 3 cm by 4 cm and which is 5 cm high. The longest length of the base is conventionally called "length" (L) and here it is 4 cm; the smallest is the “width” (l), ie 3 cm and finally, the “height” (h) is 5 cm. In summary, THE = 4, the = 3 and h = 5. #### Step 2. Focus your attention on the six faces of the cuboid

If you were to paint this box, you would have to coat 6 sides. Visualize each of them - if you can't, think of a box of cereal and look at it:

• there is a face above and a face below, both having the same surface;
• there is a front face and a back face, both having the same surface;
• there is a face to the left and a face to the right, both having the same surface;
• If you still cannot visualize the six faces, cut your box according to the edges and place them on a table, grouping them. #### Step 3. Find the surface of the underside

To begin with, we will calculate a single surface, that of the bottom face. It is therefore a rectangle, like all the other faces for that matter! This rectangle is delimited by the length and the width of the parallelepiped. To find the area of ​​a rectangle, simply multiply the two sides. Area (bottom) = length x width = Ll.

### Let's go back to our concrete example: the bottom face is 4 cm long by 3 cm wide, hence an area of ​​12 square centimeters #### Step 4. Find the surface of the top face

Wasn't it previously pointed out that this face was identical to the bottom one? Its surface is therefore equal to: Ll.

### Let's go back to our concrete example: the top face is 4 cm long by 3 cm wide, hence an area of ​​12 square centimeters #### Step 5. Find the surface of the back and front faces

Look at your sketch again and locate the front side: one side is width (w) and the other is height (h). The area of ​​this face is therefore: width x height = lh. Likewise, the area of ​​the rear face is: lh.

### Let's go back to our concrete example: l = 3 cm and h = 5 cm. The area of ​​the front face is therefore: 3 cm x 5 cm = 15 square centimeters, just as the area of ​​the back face is 15 square centimeters #### Step 6. Find the area of ​​the right and left faces

These two sides are absolutely identical. Look at your sketch again and locate the right face: one side is length (L) and the other is height (h). The area of ​​this face is therefore: length x height = Lh. The left face has the same area, Lh.

### In our example, L = 4 cm and h = 5 cm, so the area of ​​the left face is: 4 cm x 5 cm = 20 square centimeters. The right face has the same area, or 20 square centimeters #### Step 7. Add the six areas

Now that you have the areas of each of the faces, just add them to get the total area: Ll + Ll + Lh + Lh + lh + lh. You can retain this formula, which is a bit long, of course, but you will always achieve the right result.

### Part 2 of 2: Choose a simpler formula #### Step 1. Simplify the formula

You already know how to calculate the area of ​​a rectangular parallelepiped. Note that you can simplify the formula using a few simple principles of algebra. Let us take the formula found previously: area of ​​a rectangular parallelepiped = Ll + Ll + Lh + Lh + lh + lh. Some terms are identical, group them as follows:

• area of ​​the rectangular parallelepiped = 2Ll + 2Lh + 2lh. #### Step 2. Factor the number "2"

If you have learned to factorize, you already know that the formula can be simplified as follows:

• area of ​​the rectangular parallelepiped = 2Ll + 2Lh + 2lh = 2 (Ll + Lh + lh). #### Step 3. Let's see if the formula works

Let's take our example of the box 4 cm long, 3 cm wide and 5 cm high. Let's replace the letters of the formula with their respective values: