Volume • Topic 1 of 3

Volume of Rectangular Prism

What is Volume?

Volume is the amount of space inside a 3-dimensional (solid) object. Length tells you how long something is, area measures a flat surface, and volume measures how much a solid can hold.

Volume is always measured in cubic units — little cubes that are exactly 1 unit long, 1 unit wide and 1 unit tall (written cm$^3$, m$^3$, in$^3$, ft$^3$).

Measure	Dimensions	Unit	Everyday example
Length	1-D	cm	how long a pencil is
Area	2-D	cm$^2$	the top of a table
Volume	3-D	cm$^3$	water a tank holds

The Formula (and why it works)

Imagine filling a box completely with 1-cm cubes, leaving no gaps. The bottom layer is a rectangle holding $l \times w$ cubes — that is the area of the base. Stacking $h$ identical layers on top multiplies the count by $h$:

$$V = l \times w \times h$$

A second useful form is $V = (\text{area of base}) \times \text{height}$. Both give the same answer.

Special case — the cube. A cube has all edges equal, so if each edge is $a$ then $V = a \times a \times a = a^3$.

Where you meet volume in real life:

How much water a fish tank or swimming pool holds
How many small boxes fit inside a shipping carton
How much soil fills a rectangular flower bed

Common mistakes to avoid:

Writing the answer in square units (cm$^2$) instead of cubic units (cm$^3$)
Adding the dimensions ($l+w+h$) instead of multiplying them
Mixing units — always make $l$, $w$ and $h$ the same unit first

RECTANGULAR PRISM = a box. Fill it with 1-unit cubes:

        +--------------+
       /              /|
      /              / |
     +--------------+  |   height (h)
     |              |  |
     |              |  +
     |              | /
     |              |/   width (w)
     +--------------+
     <-- length (l) -->

COUNTING IN LAYERS (l = 4, w = 2, h = 3):

   Bottom layer = 4 x 2 = 8 cubes  (this is the AREA of the base)
   There are 3 such layers stacked up.
   Total cubes = 8 x 3 = 24

   So  V = l x w x h = 4 x 2 x 3 = 24 cubic units (24 cm3)

Solved Examples

Worked Example

Find the volume of a rectangular prism with length 6 cm, width 4 cm and height 5 cm.

Solution

Write the formula: $V = l \times w \times h$
Substitute: $V = 6 \times 4 \times 5$
$V = 24 \times 5 = 120$
Answer: $120$ cm$^3$

Worked Example

A fish tank is 10 inches long, 6 inches wide and 8 inches tall. Find its volume.

Solution

$V = 10 \times 6 \times 8$
$V = 60 \times 8 = 480$
Answer: $480$ in$^3$

Worked Example

Find the volume of a cube whose edge is 7 cm.

Solution

For a cube, $V = a^3 = a \times a \times a$
$V = 7 \times 7 \times 7$
$V = 49 \times 7 = 343$
Answer: $343$ cm$^3$

Worked Example

Using the base, find the volume of a prism whose base area is 24 cm$^2$ and height is 5 cm.

Solution

$V = (\text{area of base}) \times \text{height}$
$V = 24 \times 5 = 120$
Answer: $120$ cm$^3$

Worked Example

A box has volume 60 cm$^3$. Its length is 5 cm and width is 3 cm. Find its height.

Solution

$V = l \times w \times h$
$60 = 5 \times 3 \times h = 15 \times h$
$h = 60 \div 15 = 4$
Answer: $4$ cm

Worked Example

A room is 5 m long, 4 m wide and 3 m high. Find the volume of air it contains.

Solution

$V = 5 \times 4 \times 3$
$V = 20 \times 3 = 60$
Answer: $60$ m$^3$

Key Points

Volume is the space inside a 3-D shape, measured in cubic units (cm$^3$, m$^3$, ...).
Rectangular prism: $V = l \times w \times h$, or $V = (\text{base area}) \times \text{height}$.
Cube: $V = a^3$ because all edges are equal.
Multiply the three dimensions — never add them.
Make all dimensions the same unit before you calculate.
A missing dimension = volume $\div$ (product of the two known dimensions).

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.Volume is measured in:

Explanation: Cubic units (cm$^3$).

Q2.The volume of a rectangular prism is:

Explanation: Multiply all three.

Q3.Find the volume: $l=6$, $w=4$, $h=3$.

Explanation: $6\times4\times3=72$.

Q4.The volume of a cube of edge $5$ cm is:

Explanation: $5^3=125$.

Practice more MCQs → 📝 Assignment · 35 marks