## Unit Dimension and Conversion

If we analyze the concepts of dimensions and units deeply, we will find that there are many differences between them. The measurement of a physical quantity without considering the numerical values is known as the dimension. But, in the case of a unit, it can be expressed as a method to allot a measurement or number to a particular dimension. Let us analyze an illustration. Here, length has a dimension, but the measurements of the length can be expressed in the units of feet (ft) or meters (m).

In our daily usage, we find three primary unit systems.

The International System of Units (SI units),

The British Gravitational System of Units (BG Units),

The English Engineering System of Units (Also known as English units).

These three units are used for multiple purposes of measurements and are always helpful to evaluate the precise dimension of objects.

## Dimensional Formula of Volume

The dimensional formula of volume can be written as:

[M0 L3 T0]

Here,

Mass = M

Length = L

Time = T

## Dimension of Volume

We can define ‘Volume’ as the amount of space surrounded by a closed surface in a three-dimensional structure. We can derive an example such as, the substances like solid or liquid or gas, or plasma that occupy a certain space.

Mathematically, the volume is regularly measured in the SI unit known as the cubic meter. We can consider the volume of a container as the capacity or amount that it can hold all total. Such as, the quantity of fluid or gas or liquid that the container can hold. However, the volume cannot be specified for the amount of space that the container displaces itself.

The shapes that possess a three-dimensional structure are recognized to have volume, mathematically. There is a simple illustration for volume as it can be represented for the space required for certain shapes such as straight-edged, regular, and circular shapes. These shapes can be identified effortlessly by using arithmetic formulas.

Also, we can calculate the volumes of solids that do not have a commonly known regular shape. It can be done by the integral calculus for any shape if there is a formula for the boundary of the shape.

## Volume Dimensional Formula

We know that, Length x Breadth x Height = Volume

Volume can be written as,

L x L x L = L3 = Volume

So, [M0 L3 T0] = Dimensional formula of volume

m3 = SI unit of Volume

## What is the Dimensional Formula of Volume?

For a three-dimensional structure, the volume indicates the total space occupied by the same structure.

Cubic units = Measurement of the Volume

[Image will be Uploaded Soon]

If we calculate the volume of the above picture

1 unit × 1 unit × 1 unit = The volume of a unit cube = 1 unit3

unit3 is considered as one cubic unit.

In the following figure, length, breadth, and height are given for the cube.

[Image will be Uploaded Soon]

So, 1 cm × 1 cm × 1 cm = The volume of a cube with sides

V = 1 cm × 1 cm × 1 cm

= 1 cm3

cm3 is considered as one cubic centimeter.

Some significant units of conversion for volume are:

1,000 mm3 = 1cm3

1,000, 000 cm3 = 1 m3

[M0 L3 T0] = The Dimensional Formula of Volume

## How to Calculate Volume from Dimensions?

Length × breadth × height = Volume (V) ------- (1)

[M0 L1 T0] = The dimensional formula of length ------ (2)

If we substitute the equation (2) in equation (1) we will get,

Volume = Length × breadth × height

V = [M0 L1 T0] × [M0 L1 T0] × [M0 L1 T0] = [M0 L3 T0]

So, we conclude that dimensionally [M0 L3 T0] is represented for the volume.

## Solved Examples of Volume

Q1. Calculate the Volume of a Cube of Length 6 cm.

Ans: According to question,

Side of the cube = a = 5 cm

We know that,

The volume of a cube, V = a3

V = (6)3 cm3

∴ V = 216 cm3

Q2. Calculate the Volume of a Cube Having Side 8.2 cm.

Ans:

According to question,

Side of the cube = a = 8.2 cm

We know that,

The volume of a Cube, V = a3

V = (8.2)3 cm3

∴ V = 551.37 cm3

Q3. Given the Volume of a Cube is 729 cm3, then What will be the Value of the Side of that Cube?

Ans: According to question,

Volume of a cube, V = 729 cm3

We know that,

(side)3 or (length)3 = V

So, (side)3 = 729 = \[\sqrt[3]{729 cm^{3}}\] = 9 cm

We conclude that the edge of the cube is 9 cm.

Q4. What will be the Volume of the Cuboid if its Length, Breadth, and Height are 15cm, 20 cm, 25 cm Respectively?

Ans: According to question,

Length of the cuboid = 15 cm

Breadth of the cuboid = 20 cm

Height of the cuboid = 25 cm

We know that,

V= volume of the cuboid = length × breadth × height

∴ V = (15 × 20 × 25) cm³ = 7500 cm³

Q1. Can you illustrate that mL is Volume or Mass?

Ans: We recognize that gram is used for mass. For a substance, mass is its extensive property.

We know that the quantity of the substance is dependent on the extensive properties. Here, Volume is also an extensive property. It is measured in units of liters (L), milliliters (mL), or cubic centimeters (cm³).

So, mL is volume and 1 mL = 1 gm (mass).

Q2. How to Find Mass from Volume?

Ans: Density = mass / volume

We can write the above expression as, ρ = m / V

So, V = m / ρ

Yes, we can find the mass from volume. To do so, we just require to catch on the density of the specific substance that we are taking into account.

Q3. Give Your Opinion about the Variance Between Surface Area and Volume.

Ans: Surface area: It is the total of the areas of all the surfaces of a solid shape. The measurement of the surface area is in square units.

Volume: It is the product of three sides (edges) of a 3-D solid. It is measured as cubic units.

Q4. Can you Calculate the Volume of the Cylinder by Using a Sphere’s Diameter?

Ans: Yes, the calculation is as follows.

The sphere’s diameter d = 2r

So, Area of the sphere = A = 2πr²

But, in the case of the cylinder, it is slightly different.

If, h (height of the cylinder) = r (radius of the sphere)

We can calculate the volume of the cylinder: V = 2πr² h = 2πr³