The surface area of a rectangular prism is the entire space occupied by its outer layer (or faces). It is expressed in square units such as m2, cm2, mm2, and in2.
Formulas
Like all 3-dimensional shapes, we can calculate 2 types of surface areas in a rectangular prism.
Total Surface Area
The total surface area (TSA) of a rectangular prism is the sum of the lateral surface area and twice the base area. The formula to calculate the TSA of a rectangular prism is given below:
Total Surface Area of a Rectangular Prism
Lateral Surface Area
The lateral surface area (LSA) of a rectangular prism is only the sum of the surface area of all its faces except the bases. The formula to calculate the total and lateral surface area of a rectangular prism is given below:
Lateral Surface Area of a Rectangular Prism
Derivation
Now, let us derive the formulas to calculate the surface areas of a rectangular prism.
Total Surface Area
The total surface area of a rectangular prism is calculated by adding the areas of all 6 rectangular faces.
Let us consider a rectangular prism with l = length, w = width, and h = height.
Here,
The total surface area of the rectangular prism = area of [6 rectangular faces]
= area of [ABCD + EFGH + CDEF + ABGH + ADEH + BCFG]
Now,
The area of ABCD = l × w = lw
The area of EFGH = l × w = lw
The area of CDEF = w × h = wh
The area of ABGH = w × h = wh
The area of ADEH = l × h = lh
The area of BCFG = l × h = lh
Thus,
The total surface area (TSA) of the rectangular prism
= area of [ABCD + EFGH + CDEF + ABGH + ADEH + BCFG]
= lw + lw + wh + wh + lh + lh
= 2lw + 2wh + 2lh
= 2(lw + wh + lh)
Lateral Surface Area
The lateral surface area of a rectangular prism is calculated by adding the areas of all its side faces except the bases (top and bottom faces).
Thus,
The lateral surface area (LSA) of the rectangular prism
= TSA – The area of the top and bottom faces
= 2(lw + wh + lh) – area of [ABCD + EFGH]
= 2(lw + wh + lh) – 2lw
= 2(wh + lh)
= 2h(w + l)
Here are some examples to help you understand the concept better.
Solved Examples
Find the surface area of a rectangular prism given in the figure.
Solution:
As we know, Total Surface Area (TSA) = 2(lw + wh + hl) , here l = 15 m, w = 7 m, h = 5 m ∴ TSA = 2(15 × 7 + 7 × 5 + 5 × 15) = 430 cm2
Find the lateral surface area of a rectangular prism whose length is 9.5 cm, width is 8 cm, and height is 4 cm.
Solution:
As we know, Lateral Surface Area (LSA) = 2(wh + hl), here w = 8 cm, h = 4 cm, l = 9.5 cm ∴ LSA = 2{(8 × 4) + (4 × 9.5)} = 140 cm2
Find the surface area of a rectangular prism given in the figure.