# Rectangular Prism

Rectangular Prism
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Prisms are solids with identical polygon ends and flat parallelogram sides.

The different types of prisms are triangular prisms, square prisms, rectangular prisms, pentagonal prisms, hexagonal prisms, etc.

A rectangular prism is a three-dimensional shape, having six faces, and all the faces of the prism are rectangles.

Both, the base and the top of a rectangular prism must be rectangles. Also, the lateral faces are rectangles.

Let us learn more about rectangular prism, right rectangular prism, cuboids, and cubes in the concept of Rectangular Prism.

Check-out the interactive simulation to calculate the volume of a rectangular prism. Try your hand at solving a few interesting practice questions at the end of the page.

## What is Rectangular Prism?

A rectangular prism is a three-dimensional shape, having six faces, and all the faces of the prism are rectangles.

Both, the base and the top of a rectangular prism must be rectangles. Also, the lateral faces are rectangles.

Examples

• Rectangular tissue box, school notebooks, and binders, laptops, cereal boxes, fish tanks, etc.
• Large structures such as cargo containers, buildings, storage sheds, and skyscrapers.

### Right Rectangular Prism

In a right rectangular prism:

• The angles between the base and the sides are right angles.
• All its faces are rectangles.
• Each corner of the prism represents a right angle.
• Each base and top of the prism is congruent.

Think Tank
1. Is cube a rectangular prism?
2. Can you determine the volume of a rectangular prism when the area of its base and height are given but length and width are not given as separate measures?

## Rectangular Prism Properties

• A rectangular prism has 6 faces(each of them being rectangular),12 sides, and 8 vertices.
• The pairs of opposite faces of a rectangular prism are equal.
• It has a rectangular cross-section, the same cross-section along the length.
• There are two types of rectangular prisms, namely right rectangular prism and non-right rectangular prisms.

## Rectangular Prism Formula

### Rectangular Prism Surface Area

Surface area of a rectangular prism = sum of surface area of six faces = $$lw+lw+wh+wh+lh+lh$$

$$= 2(lw+wh+lh)$$

### Rectangular Prism Volume

The volume of a Rectangular Prism (V) = l × w × h

Explore the simulation given below and calculate the volume of a rectangular prism:

## How do you find the Volume of Rectangular Prism?

Have a look at the image below to see how the volume of a rectangular prism can be calculated.

How many cubic unit blocks are required to fill a rectangular prism of length 5 units, width 4 units, and height 3 units?

What is the volume of the rectangular prism?

5 blocks in each row.

4 rows in each layer.

3 layers in total.

Total number of blocks = 5 x 4 x 3 = 60

So the volume of rectangular prism = 60 cubic units.

Thus,

Volume of a Rectangular Prism (V) = l × w × h

Important Notes
1. A rectangular prism has six faces- the base, the top, and the four sides.
2. The base and top always have the same area. Similar is the case for its pairs of opposite sides.
3. Volume of Rectangular Prism: V = lwh.
4. Surface Area of Rectangular Prism: S = 2(lw + lh + wh)
5. Rectangular Prism edges = 12, faces = 6, vertices = 8

## What is the difference between Cuboid, Right Rectangular Prism and Rectangular Prism?

• A cuboid has a square cross-sectional area and a length, that is possibly different from the side of the cross-section. It has 8 vertices, 12 sides, 6 faces.
• A rectangular prism has a rectangular cross-section. It may not stand vertical, if you make it stand on the cross sectional base..
• A right rectangle prism has 6 faces, all rectangular in shape.The angles between the base and sides are right angles. Each is a right angle. Also called a cuboid as, each base and top of a rectangular prism is congruent

## Other types of Prisms

• Triangular prisms
• Square prisms
• Pentagonal prisms
• Hexagonal prisms, etc.

## Solved Examples

 Example 1

Identify the rectangular prisms from the following:

Solution

We are already familiar with the properties of a rectangular prism.

Let's just check:

•  if all six faces are rectangles.
•  if the opposite faces are equal.
•  if the cross-section along the length is the same.

Since all the above conditions are satisfied hence we can say that all three resemble rectangular prism.

 $$\therefore$$ All three resemble rectangular prism.
 Example 2

Tim wanted to add soil to his gardening bed which resembles that a shape of a rectangular prism and has the following dimensions: length = 8 ft, width = 4 ft, and height = 1 ft.

What is the maximum amount of planting soil that can be used to fill the gardening bed?

Solution

Tim's gardening bed resembles that of a rectangular prism having the following dimensions:

Length, l = 8 ft

Width, w = 4 ft

Height, h = 1 ft

The maximum amount of planting soil that can be used to fill the bed = Volume of the bed = l x w x h

That is, 8 x 4 x 1 = 32 cubic feet.

 $$\therefore$$ 32 cubic feet of soil can be used.
 Example 3

Joe has a chocolate box whose shape resembled that of a rectangular prism, the length is 6 in, height is 2 in and width is 4 in.

Find the volume of the box.

Solution

The given box has length = 6 in, height = 2 in, width = 4 in

Also, the chocolate box resembles rectangular prism so ,the volume of box = V = l x w x h.

V = l x w x h = 6 x 4 x 2 = 48 cube inches.

Volume of chocolate box = 48 cubic inches.

 $$\therefore$$ The volume of chocolate box = 48 cubic inches.
 Example 4

A middle school in Winsconsin ordered a new deep freezer, shaped as a rectangular prism, to keep it in the lunchroom to store frozen desserts.

Its dimensions are: length = 3 ft, width = 2 ft and height = 1 ft.

Find the volume of the freezer.

Solution

For the new deep freezer,

length, l = 3 ft

width, w =2 ft

height, h = 1 ft

Knowing that it is rectangular prism shaped, volume of the freezer = l x w x h

V = l x w x h = 3 x 2 x 1 = 6 cubic ft.

So the volume of the freezer is 6 cubic ft.

 $$\therefore$$ The volume of freezer is 6 cubic ft.
 Example 5

Jack's family decides to build an underground tornado shelter.

In order to fit the whole family, they need the floor to have an area of 20 ft square whereas the height should be at least 7 ft.

Noting that the shelter is more like a rectangular prism, what volume will the shelter occupy?

Solution

Area of floor = 20 square feet.

The area of the floor being rectangular in shape means l x w = 20 sq feet.

Height is given to be at least 7 ft.

And we know that volume of shelter, being a rectangular prism = l x w x h = 20 x 7 = 140 cubic ft.

So the shelter will occupy a volume of 140 cubic ft.

 $$\therefore$$ The shelter will occupy a volume of 140 cubic ft.

## Interactive Questions

Here are a few activities for you to practice.

## Let's Summarize

We hope you enjoyed learning about Rectangular Prism with the simulations and practice questions. Now you will be able to easily solve problems on the rectangular prism, right rectangular prism, cuboids, and cubes.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. What is an example of a Rectangular Prism?

A few everyday examples include tissue boxes, juice boxes, laptops, cereal boxes etc.

## 2. Is a Cuboid the same as a Rectangular Prism?

A cuboid has a square cross-sectional area and a length perhaps different from the side of the cross-section. It has 8 vertices, 12 sides, 6 faces. A rectangular prism has a rectangular cross-section.

## 3. How do you identify a Rectangular Prism?

1) See if all six faces are rectangles.
2) Check and see if the opposite faces are equal.
3) See if the cross-section along the length is the same.

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