Last modified on August 3rd, 2023

chapter outline

 

Rectangular Prism

A rectangular prism is a three-dimensional solid bounded by 6 rectangular faces, where the 2 faces are the bases (top face and bottom face), which are congruent, and the rest 4 are lateral faces. It also has 8 vertices and 12 edges.

Rectangular Prism

Due to its shape a rectangular prism is also called a cuboid. Some real life examples of rectangular prisms are a shoe box, an ice-cream bar, or a matchbox.

A rectangular prism is of 2 types based on the angle formed by the lateral faces at the bases. They are right and oblique rectangular prisms.

Right and Oblique Rectangular Prisms

Right and Oblique Rectangular Prisms
  1. Right Rectangular Prism – It has all the faces perpendicular. Thus every face is rectangular. A cuboid is a right rectangular prism as all its angles are right angles. Thus, a cuboid also holds all the above features except the oblique rectangular prism. The faces of a right rectangular prism are rectangles.
  2. Oblique Rectangular Prism – Its lateral faces are not perpendicular to its bases. So, every lateral face is parallelogram-shaped. The faces of an oblique rectangular prism are parallelograms.

Formulas

Surface Area

The surface area of a rectangular prism is the entire space occupied by its outermost layer (or faces). It is expressed in square units such as m2, cm2, mm2, and in2. We can calculate 2 types of surface areas in 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 (LSA) = 2(wh + hl), here, l = length, w = width, h = height

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 (TSA) = 2(lw + wh + hl), here l = length, w = width, h = height

Let us solve an example to understand the concept better.

Find the lateral and total surface area of a rectangular prism with length 16 cm, width  12.5 cm, and height 10 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = 2(wh + hl), here l = 16 cm, w = 12.5 cm, h = 10 cm
LSA = 2(12.5 × 10 + 10 × 16)
= 570 cm2
Total Surface Area (TSA) = 2(lw + wh + hl), here l = 16 cm, w = 12.5 cm, h = 10 cm
TSA = 2(16 × 12.5 + 12.5 × 10 + 10 × 16)
= 970 cm2

Volume

The volume of a rectangular prism is its space in the three-dimensional plane. The formula to calculate the volume of a rectangular prism is given below:

Volume (V) = l × w × h,  here l = length, w = width, h = height

Find the volume of a right rectangular prism. Dimensions of a rectangular prism are 9 cm long, 6 cm wide, and 15 cm high.

Solution:

As we know,
Volume (V) = l × w × h, here l = 9 cm, w = 6 cm, h = 15 cm
V = 9 × 6 × 15
= 810 cm2

Finding the diagonal of a rectangular prism when ALL 3 DIMENSIONS are known

Find the diagonal of a rectangular prism with dimensions of 6 cm, 5 cm, and 4 cm.

Solution:

As we know,
The equation of body diagonal of a rectangular prism is same as that of a cuboid.
∴ Diagonal (D) = ${\sqrt{l^{2}+w^{2}+h^{2}}}$, here l = 6 cm, w = 4 cm, h = 5 cm
${\therefore D=\sqrt{6^{2}+4^{2}+5^{2}}}$
= 77 cm

Last modified on August 3rd, 2023

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