# Can the areas of a circle and a square be added?

Area of a circle with radius $$r$$: $$A_c = \pi r^2$$ has the units $$[A_c]=\text{rad}\text{L}^2$$. Area of a square with side $$a$$: $$A_s = a^2$$ has the units $$[A_s]=\text{L}^2$$. Since $$[A_c] \neq [A_s]$$, we shouldn't be able to add the two.

If we are able to add $$\text{L}^2$$ and $$\text{rad}\text{L}^2$$, why aren't we able to add radians and steradians together?

As pointed out in the comments, "$$\text{rad}$$ is a unit of angle for which a circle makes an arc length equal to the radius of the circle; steradian $$\text{sr}$$ is the unit of solid angle for which an area equal to the square of the radius of sphere is projected onto the surface of the sphere. If we don't assign any units to angle, why are we not able to add radians and steradians together? $$\text{sr}$$ is effectively $$\text{rad}^2$$, I think."

This is indeed a topic of ongoing discussion amongst researchers, for (a few) references:

I could go on, but I recommend searching Google Scholar for keywords such as "Is rad an SI unit?"

As another reference, NCERT refers to $$\text{rad}$$ and $$\text{sr}$$ as units in "Units and Dimensions" (Chapter 2, Class XI physics textbook), albeit dimensionless ones (which is also a topic of ongoing discussion.

• Quite confusing : If $r$ is in , say , $m$ , then both areas are in $m^2$ , hence can be added. What do you mean with "radL^2" ? Commented Dec 13, 2022 at 12:00
• @Peter the unit of $\pi$ is $\text{rad}$. Commented Dec 13, 2022 at 12:02
• $\pi$ is a constant and has no unit. Neither is rad a unit. It tells us the ratio of the arc length in a unit circle and its radius. Commented Dec 13, 2022 at 12:07
• @Peter $\text{rad}$ is a unit of angle for which a circle makes an arc length equal to the radius of the circle; steradian $\text{sr}$ is the unit of solid angle for which an area equal to the square of the radius of sphere is projected onto the surface of the sphere. If we don't assign any units to angle, why are we not able to add radians and steradians together? $\text{sr}$ is effectively $\text{rad}^2$, I think. Commented Dec 13, 2022 at 12:11
• "why are we not able to add radians and steradians" $\;-\;$ Simply because dimension compatibility does not mean that you can arbitrarily compare/add/subtract them. $\;$ "This is indeed a topic of ongoing discussion" $\;-\;$ No, it's not. Please quote relevant sources and very specific assertions if you wish to support yours.
– dxiv
Commented Apr 18, 2023 at 7:45

I would like to paint to surfaces: a (piece of) a circle and a square. How many liters of paint do I need to buy in order to cover the entire surface?

As both surfaces have different measurement units, you can't cover both surfaces using paint from the same container, so you need at least to buy two paint containers.

Do you see the problem with your approach?

Firstly, the $$\pi$$ used in the area of a circle is used as a constant like the number 1 and 2 rather than the angle (such as $$\theta^c$$ or $$\theta^{\circ}$$). I see that you have asked certain question on thermodynamics on the chemistry site, then tell me one thing that you must have studied. If the function ln(x) can only intake numerical values of x, how do we say $$\Delta G^0 = RTln(K)$$ when the equilibrium constants K can have dimensions? That is because we use the "numerical value" of the equilibrium constant (expressed in atm and molarity) rather than simply the equilibrium constants with the dimensions. This is exactly what we did here, we used the numerical value of $$\pi = 3.141...$$ rather than $$\pi ^ c$$ or $$\pi^{\circ}$$.

Secondly, there is not much of a raging debate or research among scholars and in the literal SI official brochure, it is mentioned that the units of radian and steradian can be used in SI units as and how required (according to the convenience of the author). Since the definition of radian and steradian is such that they have to become dimensionless. Note that they are not "unitless", they are dimensionless meaning they can not be represented using the 7 fundamental quantities because of the fact that radian is defined as $$\frac{arc\;length}{radius}$$ and steradian is defined as $$\frac{area}{radius^2}$$. This means their dimensions are non existent but they are "units" because they are literally defined as given above (the definition of unit is something used to measure a quantity or how much some measured quantity is with respect to that unit and so we can measure angle using radian, solid angle using steradian).

Also, we cannot add dimensionless quantities together just becuase they are dimensionless. They still have units which are different (in your case radian and steradian). Like for example, we can define infinitely many units without dimensions such as let us call this unit $$\xi = \frac{force}{(rate\;of\;change\;mass) \cdot area}$$ but that does not mean you can add this quantity and radian, becuase although as per certain rules, you can only add same dimension quantities together, you can also not add two quantities that differ fundamentally from each other or which have different "units". For example : can you add angular velocity $$\omega$$ and $$\frac{1}{time}$$ just because they have the same "dimensions"? No you cannot and that is actually the sole reason why we can write $$\omega$$ as radian per second rather than only persecond (as stated by the official SI brochure) because when you write this unit you realise that $$\omega$$ and $$\frac{1}{time}$$ differ fundamentally or in the case of "units".

P.S. : Kindly do not cite NCERT and other local books because we all know how flawed NCERT is in half of the places.

• I think the last sentence in the quoted NCERT passage does say that NCERT knows what's being talked about. Commented Nov 25, 2023 at 16:38
• AFAIK, K in that particular formula for calculating Gibbs energy is dimensionless by definition. Commented Nov 25, 2023 at 16:50
• Isn't that the point I am trying to make? K is used in that formula as dimensionless, but is it actually? No, we use the numerical dimensionless value, just as of $\pi$ in the area of the circle Commented Nov 25, 2023 at 17:54
• No, we don't just use the numerical value. The definition of K is dimensionless. Commented Nov 25, 2023 at 17:58
• Uhm what? Haven't you seen the formula for K of a reaction, let's say $N_2 + 3H_2 \rightarrow 2NH_3$ (all in gaseuous form). Isn't the unit of k here $\frac{1}{(atm)^2}$? Commented Nov 26, 2023 at 13:37