Dimensional Formula of Area

Physical quantities are related to the dimensions of the measurement units used to define them. This helps us perform mathematical calculations that are easier, more precise, and quicker. In other words, Dimensional analysis is the study of dimensional formulae. It is the technique used to manipulate dimensional formulae.

Dimensional formula: 

In terms of dimensions, a dimensional formula is an equation that expresses the relationship between fundamental and derived units (equation). The letters L, M, and T represent the mechanic’s three basic dimensions: length, mass, and time. All physical quantities can be stated in terms of the fundamental (base) units of length, mass, and time, multiplied by some factor (exponent).

• The dimension of the amount in that base is the exponent of a base quantity that enters into the expression.

• The units of fundamental quantities are expressed as follows to determine the dimensions of physical quantities:

‘L’ stands for length,

‘M’ for mass, and

‘T’ for time.

Example: The area is equal to the product of two lengths. As a result, [A] = [L2]. An area has two dimensions of length and zero dimensions of mass and time. In the same way, the volume is the product of three lengths. As a result, [V] = [L3]. The volume dimension has three dimensions: length, mass, and time.

Dimensions in units and measurements

The dimensions can be written as the powers of the fundamental units of length, mass, and time. It depicts their nature and does not shows their magnitude.

Dimensional Formula of Area

Area of the rectangle = length x breadth

= l x l (where breadth is also showing the length of the side)

= [L1] X [L1]

= [L2]

Here, we can see the length to the power of 2, and we cannot find the dimension of mass and time. Hence, the dimension of the area of a rectangle is written as [M0 L2 T0]

Dimensional formula dimensional equation

The dimensional formula depicts the dependency of physical quantity with fundamental physical quantity and the powers.

Example

Let us take the formula of speed.

Speed = Distance / Time

Therefore, the distance can be written in length [L]

Time can be written as [T]

The dimensional formula would be [ M0 L1 T-1]

Hence, we can conclude that the speed is dependent on only length and time, not mass.

Dimensional equation

So, to get the dimensional equation, the physical quantity is equated with the dimensional formula.

Example

Velocity = [ M0 L1 T-1]

Uses of Dimensional Formula:

1. It can be a helpful tool to check the consistency and coherency of a dimensional equation.

2. The dimensional formula will be used to establish the correlation between the physical quantities of physical phenomena.

3. These formulas can change the units from one system to another.

Limitations of dimensional formulas

• It is not concerned with the dimensional constant.

• The formula that contains functions like trigonometric, exponential, logarithmic and the like cannot be derived.

• In the context of a physical quantity being a scalar or vector, it does not provide any information about whether the quantity is physical or not.

The surface area of the cuboid

The primary example of cuboids is brick, matchbox and chalk box. The figure shows the cuboid’s length, breadth, and height. Let the vertices be ABCDEFGH with six rectangular faces.

The surface area of a solid figure is the measurement of the sum of the areas of all its six faces. 

Here, the surface can be calculated as

lxb + bxh + hxl + lxb + bxh + hxl

= 2( lb + bh + hl)

Now, draw the line which joins BE and EC,

BE2 = AB2 + AE2    

( As EAB = 900)

Or BE2 = l2 + b2 —-(1)

Also, EC2 = BC2 + BE2 

( As CBE = 900)

Or EC2 = h2 + l2 + b2

So, EC = h2 + l2 + b2

Hence, diagonal of a cuboid = h2 + l2 + b2

Conclusion

The dimension is the property of units and measurements used for many applications like correction, comparison, and derivation of the physical quantity. The dimensional formula has the basic terms of length, mass, and time. The equation of physical quantity and dimension is called the dimensional equation. We have also covered the characteristics of Dimensional Formula & Equations, including the principle of homogeneity of dimensions. Finally, the dimensional analysis is explained.