Surface Area of Cuboid

A cuboid is a three-dimensional shape with six rectangular faces: four lateral faces and two faces at the top and bottom. The surface area is the total area of its six rectangular faces, calculated based on its dimensions. The surface area of a cuboid is measured in square units because it represents the total area of all six of its faces.

There are two types of surface area of a cuboid. The surface area of a cuboid is calculated by using its length, breadth, and height — it's all based on its dimensions. Here are the types:

1. Total surface area
    
2. Curved surface area 

Total surface area 

A cuboid has 6 rectangular faces, and the total surface area is the combined area of all those faces. To find it, we calculate the area of each face and then add them all together.
 

The image shows the areas of all six rectangular faces of the cuboid

There are 2 faces of length × breadth, 2 faces of breadth × height, and 2 faces of height × length. So the total surface area is:

Total surface of area = 2 (l × b) + 2 (b × h) + 2 (l × h)
                                 = 2 (lb + bh + hl)

Lateral surface area 

The total surface area and the lateral (or curved) surface area of a cuboid are different. The total surface area includes all six faces: top, bottom, front, back, left, and right. The lateral surface area includes only the four side faces — it does not include the top and bottom.

We already know:

Total Surface Area (TSA)=2(lb + bh + hl)

This includes the top and bottom faces (ABCD and EFGH), which are the same area (l  b). The Lateral Surface Area (LSA) is the area of the opposite faces only — that is, without the top and bottom.

So we subtract the area of the top and bottom:

Area of ABCD = l × b

Area of EFGH = l × b

So combined = 2 × lb 

Curved Surface Area = Total Surface Area −2(lb)

Curved Surface Area = 2(lb + bh + hl) − 2(lb)

Now subtract:

= 2(bh + hl)

Factor out 2h:

Curved Surface Area of a Cuboid = 2h(b + l)

The total surface area of a cuboid includes all six faces — top, bottom, front, back, left, and right. The lateral surface area (also called curved surface area) includes only the four vertical side faces — it does not include the top and bottom. Let’s see the difference.

The surface area of a cuboid is the total area that covers the outside of the cuboid, including all six rectangular faces — the top, bottom, front, back, and the two sides. These steps will help you calculate the surface area of a cuboid:

Find the lateral surface area and the total surface area of a cuboid with the following dimensions: Length = 10 cm, Breadth = 6 cm, Height = 4 cm.

1. Total surface area of a cuboid 

TSA = 2(lb + bh + hl)

Step 1: Substitute the length, breadth, and height in the formula
 = 2(10 ×  6 + 6 × 4 + 4 × 10)

Step 2: Calculate the value

= 2 (60 + 24 + 40)

= 2(124)

TSA = 248 cm²

2. Curved surface area of a cuboid

CSA = 2h(l + b)

Step 1: Substitute the length, breadth, and height in the formula
= 2 × 4 (10 + 6)

Step 2: Calculate the value

= 8 × 16

CSA =128 cm²

We use the surface area of a cuboid in many everyday situations—often without even realizing it. Here are some simple examples you might come across in real life:

Painting boxes:  If you're painting a cupboard, wooden box, or shipping container, you need to know the total surface area. That way, you can buy just the right amount of paint. 

Wrapping a Gift Box: Ever wrapped a gift and run out of paper? Knowing the surface area of the box helps you cut the right amount of wrapping paper to cover the entire box neatly.

Packaging Design: Some companies use surface area to design flat cardboard pieces that are folded into product boxes or cartons. This helps them cut out the correct size before folding it into shape.

Making Labels: If you're making a label that sticks around the sides of a box, you need the lateral surface area. It tells you how big the label should be to fit perfectly.

Construction Work: Builders often work with cuboid-shaped rooms or structures. They use surface area to figure out how much paint, tiles, or wallpaper they’ll need for the walls, floors, or ceilings.