Surface Area of Cubes and Rectangular Prisms

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How do you find the total surface area of a rectangular box?

A rectangular prism where all the sides are the same is a cube.

The surface area of a prism is the total sum of the areas of all its side faces. Representing a solid in two-dimensional form is called a net.

Surface area is the total area of all surfaces on a 3dimensional shape.

Cuboids or rectangular prisms have 6 rectangular faces.

We mark the length l, width w, and height h of the prism and use the formula,  

Area of top and bottom face = 2 lw

Area of right and left side face = 2hw

Area of front and back face = 2hl

Example: Find the surface area of prism shown in the figure.

S = 2lw + 2lh + 2wh

= 2(3)(5) + 2(3)(6) + 2(5)(6)

= 30 + 36 + 60

= 126 square inches

Example: Find the surface area of an aquarium of dimensions 10 cm, 8 cm and 5 cm.

Formula for Surface area = 2lw + 2lh + 2wh

= 2.10.5 + 2.10.8 + 2.8.5

= 340 cm²

How do you find the surface area of a cube?

When all sides in a rectangular prism are equal, it is a cube. So, each face is a square.

A cube has 6 square faces.

In a cube all sides are equal so we calculate the surface area by 6 × side²

Example: Calculate the surface area of a cube of side 4 mm

Surface area of a cube = 6 × side² 

= 6 × 4²

= 96 mm²

 

Check point   
  1. Find the surface area of a rectangular wooden TV unit of length 18 cm, width 5 cm, height, 6 cm.
  2. Find the surface area of this cube.

  1. Aurora wants to calculate the surface area of a carton box which she needs to wrap. Length of the box is 7 cm, width is 5 cm, and 2 cm is the height.

  1. Find the surface area of a rectangular tube if the length, width, and height are as follows:

  1. Which of the following ice cubes will melt faster? “A” which measures 3 mm, 2 mm, 1 mm or “B” the other measures 2 mm, 1.5 mm, 1.3 mm. Calculate the surface area for both (Hint: the more is the surface area, the quicker it melts).

  1. What will be the surface area of cube with side 32 inches?
Answer key
  1. 456 cm²
  2. 726 mm²
  3. 118 cm²
  4. 76 square inches
  5. A = 22 cm², B = 15.1 cm² A is the ice cube which will melt faster, due to larger surface area.
  6. 6,144 square inches