# 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.

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

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

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