What is the Formula for the Surface Area of a Rectangular Prism?
A rectangular prism is a three dimensional shape with 4 rectangular faces and 2 rectangular bases. The four rectangular faces of the rectangular prism are known as the lateral faces and the top and bottom rectangular faces are known as the base of the prism. The surface area of a rectangular prism can be calculated by adding up the area of the six rectangular faces that include the lateral faces and the two bases.
Formula for the Surface Area of a Rectangular Prism
Surface area of a rectangular prism = Sum of the area of all six faces
= 2lb + 2hl + 2bh
= 2 (lb + bh + hl)
Where,
l = base length of the rectangular prism
b = base width of the rectangular prism
h = height of the rectangular prism
Solved Examples
Example 1: Tom has built his own model building in the shape of a rectangular prism. The dimensions of the model are: base length is 20 inches, width of the base is 18 inches, and the height of the building is 5 inches. Find the surface area of the building. (Thickness of the model is considered as negligible).
Solution:
As stated in the question,
Base length of the rectangular prism, l = 20 in
Base width of the rectangular prism, b = 18 in
Height of the rectangular prism, h = 5 in
Surface area of the model = 2 (lb + bh + hl)
= 2 (20 x 18 + 18 x 5 + 5 x 20) [Substitute the values]
= 2 (360 + 90 + 100) [Apply PEMDAS rule]
= 2 (550) [Add]
= 1,110 \( in^2\)
So, the surface area of the model is 1110 square inches.
Example 2: Find the height of the rectangular prism that has a base length of 5 m, base width is 4 m, and surface area is 76 \( m^2\).
Solution:
As stated in the question,
Surface area of the rectangular prism, S = 76 \( m^2\).
Base length of the rectangular prism, l = 5 m
Base width of the rectangular prism, b = 4 m
Surface area of the rectangular prism = 2 (lb + bh + hl)
76 = 2 (5 × 4 + 4 × h + 5 × h ) [Substitute the values]
38 = 5 × 4 + 4 × h + 5 × h [Divide both sides by 2]
38 = 20 + 9 × h [Add the variable]
18 = 9 × h [Subtract 20 from both sides]
2 = h [Divide both sides by 2]
Therefore, h = 2 meters.
So, the height of the prism is 2 meters.
Example 3: John wants to give a laptop to her mother. The dimension of the laptop box is 15 inches by 12 inches by 3 inches. He wants to wrap the box with gift wrapping paper. Find the surface area of the wrapping paper that is required to wrap the box.
Solution:
Surface area of the wrapping paper = Surface area of the laptop box
= 2 (lb + bh + hl)
= 2 (15 × 12 + 12 × 3 + 3 × 15) [Substitute the value]
= 2 (180 + 36 + 45) [Apply PEMDAS rule]
= 2 (261)
= 522 \( in^2\)
So, the required area of the wrapping paper is 522 square inches.
There are various objects around us that are in the shape of a rectangular prism. Some of them are, rectangular glass prisms, rectangular pieces of cheese and butter, books, boxes, buildings, bricks, mobiles and laptops.
There are six rectangular faces, 4 rectangular faces are the lateral faces and two rectangular faces on the top and the bottom creating a rectangular prism.
Rectangular prisms have 12 edges and 8 vertices.
Units to be used to represent the surface area of a rectangular prism are square meters, square centimeters, square inches, square feet and other square units of length.