Surface Area of a Cuboid
Summary
A (rectangular) cuboid is a closed box which comprises of 3 pairs of rectangular faces that are parallel to each other, and joined at right angles. It is also known as a right rectangular prism. It has 8 vertices, 6 faces, and 12 edges. The cube is a cuboid whose faces are all squares.
Imgur
imgur
The figure above shows the developmental figure of a \(a\times b\times c\) cuboid. Notice that the developmental figure consists of two \(a\times b\) faces, two \(b\times c\) faces, and two \(c\times a\) faces. The surface area of a cuboid is the sum of the areas of these 6 faces. Hence the surface area of a \(a\times b\times c\) cuboid is \(2(ab+bc+ca)\).
Examples
What is the surface area of a \( 2 \times 3 \times 4 \) cuboid?
The surface area is \( 2 ( 2 \times 3 + 3 \times 4 + 4 \times 2) = 52 \). \( _\square \)
hi
There is a cuboid whose base is a square and height is 10 cm. If the area of one side face is \(120\text{ cm}^2\), what is the total surface area of the cuboid?
Since the area of one side face is \(120\text{ cm}^2\), the side length of the square base is \(\frac{120}{10} = 12\text{ cm}\). Thus, the cuboid is a \(12\text{ cm}\times 12\text{ cm}\times 10\text{ cm}\) cuboid. Hence, the surface area of the cuboid is
\[2 \times (12 \times 12 + 12 \times 10 + 12 \times 10) = 768\ (\text{cm}^2).\ _\square \]
A cuboid's base has an area of \(20\text{ cm}^2\), and a perimeter of 20 cm. If the cuboid's height is 6 cm, what is the surface area?
The sum of the areas of the side faces is (perimeter of base) \(\times\) (height), and the surface area of the cuboid is (sum of areas of the side faces) + 2 \(\times\) (area of base). Thus, the surface area of the cuboid is \[20 \times 6 + 2\times 20 = 160\ (\text{cm}^2).\ _\square \]
Consider a cuboid whose base is a \(3\text{ cm}\times4\text{ cm}\) rectangle. If the volume of the cuboid is \(60\text{ cm}^3,\) what is the surface area?
An \(a \times b \times c \) cuboid has a volume of \( abc \) and a surface area of \( 2 ( ab+bc+ca) \). Since the cuboid's volume is \(60\text{ cm}^3\) and the base is a \(3\text{ cm}\times4\text{ cm}\) rectangle, the height is \(\frac{60}{3 \times 4} = 5\text{ cm}.\) Then, the surface area of the cuboid is \[2 \times ( 3 \times 4 + 3 \times 5 + 4 \times 5) = 94\ (\text{cm}^2).\ _\square \]