Volume of a Cuboid and Cube

🏆 Formula to Calculate the Volume of a Cuboid

$$ \text{Volume} = l \times w \times h $$

Where:

• • $l$ is the length
  • $w$ is the width
  • $h$ is the height

Example: What is the volume of a cuboid with:

• • Length = $5\,\text{cm}$
  • Width = $3\,\text{cm}$
  • Height = $2\,\text{cm}$

$$
V = 5\,\text{cm} \times 3\,\text{cm} \times 2\,\text{cm} = 30\,\text{cm}^3
$$

So the volume of this cuboid is 30 cubic centimeters.

🏆 Formula to Calculate the Volume of a Cube

A cube is a special type of cuboid where the length, width, and height are all equal.

$$ \text{Volume} = s \times s \times s $$

Where:

• • $s$ is the length of each side

Example: What is the volume of a cube with:

• • Side length = $5 \text{cm}$

$$ V = 5\,\text{cm} \times 5\,\text{cm} \times 5\,\text{cm} = 125\,\text{cm}^3$$

So the volume of this cube is 125 cubic centimeters.

You can rearrange the volume formula to find a missing dimension.

For example:

• • Volume = $120 \text{m}^3 $
  • Length $ l= 10 \text{m} $
  • Height $ h= 2 \text{m} $
  • We want to know the Width $w$

Step 1: Plug in the known values

$$
120\,\text{m}^3 = 10\,\text{m} \times w \times 2\,\text{m}
$$

Step 2: Multiply the known values

$$
120\,\text{m}^3 = 20\,\text{m}^2 \times w
$$

Step 3: Solve for the width

$$
w = \frac{120\,\text{m}^3}{20\,\text{m}^2} = 6\,\text{m}
$$

So the width of this cuboid is 6 meters.

Method: To find the volume of a complex solid, break it down into smaller cuboids and find the volume of each one. Then add the volumes together.

Example: A complex 3d shape can be split into 2 cuboids.

• For Cuboid A:

$$V = 1\,\text{cm} \times 2\,\text{cm} \times 3\,\text{cm} = 6\,\text{cm}^3$$

• For Cuboid B:

$$V = 2\,\text{cm} \times 2\,\text{cm} \times 1\,\text{cm} = 4\,\text{cm}^3$$

• Adding both together:

$$V = 6\,\text{cm}^3 + 4\,\text{cm}^3 = 10\,\text{cm}^3$$

So the volume of the complex solid is 10 cubic centimeters.

Question: If a cuboid-shaped container is being filled with water, and you know the dimensions and the filling speed, you can calculate how long it takes to fill.

• • Length = $50\,\text{cm}$
  • Width = $30\,\text{cm}$
  • Height = $20\,\text{cm}$

Step 1: Calculate the volume of the container

$$ V = 50\,\text{cm} \times 30\,\text{cm} \times 20\,\text{cm} = 30{,}000\,\text{cm}^3$$

Step 2: Convert cubic centimetres to litres

$$ 1{,}000\,\text{cm}^3 = 1\,\text{L} \quad \Rightarrow \quad 30{,}000\,\text{cm}^3 = 30\,\text{L}$$

Step 3: Use the filling speed to find the time

Filling speed = $10,\text{L/min}$

$$ \text{Time} = \frac{30\,\text{L}}{10\,\text{L/min}} = 3\,\text{minutes}$$

Answer: So, it takes 3 minutes to fill the container.