Surface Area of a Cuboid



In this lesson we will learn

  • how to calculate the surface area of a cuboid.
  • how to solve word problems about cuboids.
  • how to use the net of a cuboid to calculate the surface area.

Related Topics: More Geometry Lessons

Surface Area of a Cuboid

A cuboid is a 3-dimensional object with six rectangular faces. All its angles are right angles and opposite faces are equal. A cuboid is also called a rectangular prism or a rectangular solid.

In a cuboid, the length, width and height may be of different lengths. A cube is a special case of a cuboid in which all six faces are squares.

rectangular solid



To calculate the surface area of the cuboid we need to first calculate the area of each face and the add up all the areas to get the total surface area.

Total area of top and bottom surfaces is lw + lw = 2lw
Total area of front and back surfaces is lh + lh = 2lh
Total area of the two side surfaces is wh + wh = 2wh

Surface area of cuboid = 2lw + 2lh + 2wh = 2(lw + lh + wh)

Volume of rectangular prism = lwh

Worksheet to calculate the volume and surface area of a cuboid.



Example

Find the surface area of the following cuboid.

cuboid

Solution:

Total area of top and bottom surfaces is 2 × 5 × 6 = 60 in2
Total area of front and back surfaces is 2 × 5 × 3 = 30 in2
Total area of the two side surfaces is 2 × 6 × 3 = 36 in2

Surface area of cuboid = 60 + 30 + 36 = 126 in2

or

l = 6 in, w = 5 in and h = 3 in
Surface area of cuboid = 2(lw + lh + wh) = 2 (6 × 5 + 6 × 3 + 5 × 3) = 126 in2



The following video show how to derive the formula of the surface area of a cuboid.



The following videos show how to find the surface area of a cuboid.







Surface Area of Cuboids






Word problems about cuboids

This video shows how to solve problems about the volume and surface area of a cuboid.

Problem: Cereal boxes have dimensions as shown in the figure. The cereal box factory is about to close for the weekend, and there are just 1000 in2 of cardboard left. How many more cereal boxes can be produced at the factory with the remaining cardboard?



Problem: On Southwest Airlines, the maximum size of a carryon bag is a length of 24 in., a width of 10 in., and a height of 16 in. How much can be packed in this maximum sized bag?

Problem: A seamstress needs to cover a box that is 8 cm long, 5 cm high and 4 cm wide with material on all sides. How much material does she need?



This video shows a problem which compares the surface area of a cube and a cuboid.







Nets of a Cuboid

Another way to look at the surface area of a cuboid is to consider a net of the cuboid. The net is a 2-dimensional figure that can be folded to form a 3-dimensional object.

Imagine making cuts along some edges of a cuboid and opening it up to form a plane figure. The plane figure is called the net of the cuboid.

The following net can be folded along the dotted lines to form the cuboid.

net of cuboid

We can then calculate the area of each rectangle in the net and add them up to get the surface area of the cuboid.

This videos show how to find the surface area of a rectangular prism using nets.









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