A rectangular prism is a three-dimensional solid shape with six faces that including rectangular bases.

A cuboid is also a rectangular prism. The cross-section of a cuboid and a rectangular prism is the same.

## Real-Life Examples

A pencil box and a book are rectangular prisms. It may surprise you, but your room is also a cuboid or a rectangular prism.

## Properties of a Rectangular Prism:

- A rectangular prism has 8 vertices, 12 sides and 6 rectangular faces.
- All the opposite faces of a rectangular prism are congruent.
- A rectangular prism has a rectangular cross section.

## Types of Rectangular Prisms

There are two types of rectangular prisms:

**Right Rectangular Prism:**The angle formed by the faces with any of its bases is 90° or a right angle. All the faces, including the lateral ones, are rectangular.**Non-Right (Oblique) Rectangular Prism:**The faces of such a prism are not at right angles to the bases. The shape of each face is more like a parallelogram than a perfect rectangle.

## Surface Area and Volume of a Rectangular Prism:

The rectangular prism has three dimensions. It means that it has a surface area and volume.

Since all the faces of a rectangular prism are rectangles and opposite faces are equal, the surface area of a rectangular prism can be calculated using the following formula:

Total Surface Area $= 2 {(\text{width} \times \text{length} ) + (\text{length} \times \text{height}) + (\text{width} \times \text{height})}$

Lateral Surface Area $= 2 {( \text{length} \times \text{height} ) + (\text{width} \times \text{height} )}$

Volume of a rectangular prism is simply obtained by multiplying all three dimensions $–$ length, height and width.

Volume $= \text{length} \times \text{width} \times \text{height}$

## Fun Facts

– Rectangular prisms are the most commonly used prisms in real life, especially in packaging, from cereal boxes to cartons and parcels delivered by mail.

## Solved Examples

**Find the volume of a right rectangular prism whose length**$= 8$**cm, width**$= 5$**cm, and height**$= 16$**cm.**

Volume $=$ (LWH)

$= 8516$

$= 640$ cm^{3}

**What is the total surface area of a right rectangular prism with length**$= 5$**feet, width**$= 4$**feet, and height**$= 6$**feet.**

Total surface area $= 2 { (\text{WL})+(\text{LH})+(\text{WH})}$

Total surface area $= 2 {(45)+(56)+(46)}$

$= 2(20+30+24)$

$= 148$ ft^{2}

**The dimensions of a rectangular prism are length = 2.5 cm, width = 4.5 cm, and height = 1.5 cm. Find the volume.**

Volume $=$ LWH

$= 2.5 4.5 1.5$

$= 16.875$ cm^{3}

**For a rectangular prism, length**$= 4$**feet, width**$= 4$**feet and height**$= 9$**feet. Find the lateral surface area.**

Lateral surface area $= 2 { (\text{LH})+(\text{WH})}$

Lateral surface area $= 2 { (49)+(49)}$

$= 2 ( 36+36 ) = 144$ ft^{2}

## Practice Problems

## What is a Rectangular Prism? Definition With Examples

### What is the volume of a rectangular prism whose dimensions are length $= 2$ cm, width $= 2$ cm, and height $= 4$ cm?

Volume $= 2 \times 2 \times 4 = 16 \text{cm}^3$

### The volume of a rectangular prism is $50 \text{cm}^3$. Also, length $= 2$ cm and width $= 5$ cm. What is its height?

$h$ $= \frac{V}{(l \times w)} =\frac{50}{(2\times5)}=5$ cm

### The dimensions of a rectangular prism are length $= 4$ cm, width $= 6$ cm and height $= 10$ cm. What is the total surface area?

Total Surface Area $=$ $2{(4\times6)+(6\times10)+(10\times4)} =248 \text{cm}^2$

### The dimensions of a rectangular prism are length $= 5$ cm, width $= 6$ cm and height $= 7$ cm. What is the lateral surface area?

Lateral Surface Area $=$ $2\times{ (5\times7)+(6\times7) } = 154 \text{cm}^2$

## Frequently Asked Questions

**How many bases does a rectangular prism have?**

A rectangular prism has two congruent and parallel rectangular bases.

**What is the net of a rectangular prism used for?**

The net of the rectangular prism is used to find the total surface area.

**Can a right rectangular prism have a square face?**

Yes. Some prisms are named according to special properties. A right rectangular prism with a square base is called a right square prism.

**What makes a right rectangular prism different from an oblique rectangular prism?**

The bases of the right rectangular prism are perpendicular to each other. Whereas the bases of the oblique rectangular prism are not perpendicular.