Dimensions of Area

We can obtain the dimension of any physical quantity from the fundamental units. By fundamental units, we mean the units of mass, length, and time. Here, our focus is on finding the dimension of the area.

In dimensional analysis, for finding out the dimension of any physical quantity, whether it is a force, acceleration, or many more; we represent mass by [M], length by [L], and time by [T].

We know that area of any geometric shape, let’s say, the square is the square of its side. The side is the length, so the dimension for length is [L], and the square of this length will be [L²].

So, the dimensional formula of the area of the square is [M⁰L²T⁰].

Two Methods to Write a Dimensional Formula

There are two tricks to write the dimensional formula. They are:

1. Using formula

2. Using units 

Let’s understand by an example:

We know that the surface area of the sphere is 4 * π * r² . . .(1) and the unit of r is meter.

So, r*r becomes m².

We used the formula and got the unit m².

Now, we know that m (meter) is the unit of length and the dimension of length is [L], and

The dimensions of constant values, 4 and π are \[[M^{0}L^{0}T^{0}]\].

We know that when two similar quantities with different powers are multiplied, then the two powers are added. Let’s see an example of the same:

\[a^{b} \times a^{c} = a^{b + c}\]

Let’s apply this concept in equation (1):\[[M^{0}L^{0}T^{0}] \times [M^{0}L^{0}T^{0}] \times [M^{0}L^{2}T^{0}]\]

Now adding the powers on common terms:

\[[M^{0 + 0 + 0} L^{0 + 0 + 2} T^{0 + 0 + 0}] = [M^{0} L^{2} T^{0}]\] is the dimensional formula for the surface area of the sphere.

Let’s do the derivation of dimensions of area for different geometric shapes:

Dimensional Formula of the Area for Different Shapes

Let’s find the dimensional formula of area for different geometrical shapes:

1. Rectangle

We know that area of the rectangle = L x B

⇒ The dimension of L and B  = [L]

So, the dimensional formula for the area of the rectangle = \[[M^{0} L^{2} T^{0}]\]

Total surface area of the rectangle = 2 * (l*b + b*h + h*l) 

The dimension of l = b = h = [L]

So, \[2 \times ([L^{2}] + [L^{2}] + [L^{2}]) = 6 \times [L^{2}]\]

The dimension of 6 (a constant value) = \[[M^{0} L^{0} T^{0}]\]

= \[6 \times [L^{2}] = [M^{0} L^{0} T^{0}] \times [M^{0}L^{0} T^{0}]\] . . .(2)

Now, adding the powers of the term, as shown below:

= \[[M^{0 + 0}L^{0 + 2} T^{0 + 0}]\]

We get the dimensional formula for the total surface area of the rectangle is:

\[[M^{0} L^{2} T^{0}]\]

2. Circle

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We know that area of the circle is πr²

Here, the dimension of radius r is [L]

The dimension formula of \[r^{2} = [M^{0} L^{2} T^{0}]\]

π is constant, so its dimension is \[[M^{0} L^{0} T^{0}]\]

Now, using the formula = \[\pi r^{2} = [M^{0} L^{0} T^{0}] \times [M^{0} L^{0} T^{0}]\]

We know that in multiplication, we add the powers in similar terms.

Let’s add the powers now:

\[[M^{0 + 0} L^{0 + 2} T^{0 + 0}]\]

We get the dimension formula for the area of the circle as \[[M^{0}L^{2}T^{0}]\]. 

3. Sector

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The area of a sector is \[\frac{\theta}{360^{\circ}} \pi r^{2}\] ...(3)

The unit of θ is radian, whose dimension is \[[M^{0}L^{0}T^{0}]\] ….(4)

Similarly, for the constant values, i.e., 360° and π, the dimension is:

\[[M^{0} L^{0} T^{0}]\]….(5)

The dimension of \[r^{2} = [M^{0} L^{2} T^{0}]\]...(6)

On putting the value of (4), (5), and  (6) in equation (3), we get

= \[\frac{[M^{0}L^{0}T^{0}]}{[M^{0}L^{0}T^{0}]} [M^{0}L^{0}T^{0}] \times [M^{0} L^{2} T^{0}]\] …(7)

We know that the powers in the common terms in the division form are subtracted, let’s use the formula mentioned below:

= Using the formula \[\frac{a^{b}}{a^{c}} = a^{b - c}\] and \[a^{b} \times a^{c} = a^{(b + c)}\] we get:

= \[[M^{0 - 0}L^{0 - 0} T^{0 - 0}] \times [M^{0 + 0} L^{2 + 0} T^{0 + 0}] = 1 \times [M^{0} L^{2} T^{0}]\]

So, the dimensional formula for the area of a sector is \[[M^{0} L^{2} T^{0}]\].

4. Ellipse

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The area of the ellipse is π *a * b.

Here, a and b are the lengths of the major and the minor axis, respectively.  The dimensional formula for each of these is [L].

So, for a * b, the dimensional formula will be \[[M^{0} L^{2} T^{0}]\].

We know for π, the formula is \[[M^{0}L^{0}T^{0}]\].

π *a * b = \[[M^{0}L^{0}T^{0}] \times [M^{0}L^{2}T^{0}]\]

So, the dimensional formula for the area of an ellipse = \[[M^{0}L^{2}T^{0}]\].