The magnitude of torque is defined as the product of the perpendicular (to the object) component of the force I apply and the distance between the axis of rotation and the point of application of the force.
Now, this is a definition, and it does not have a proof. However, it is motivated through the fact that $\tau\propto\text{F}$ and $\tau\propto\text{r}$. Now comes my question: why is a constant of proportionality not included in the equation for torque?
The equation for torque (τ) can be expressed as:
Τ = r × F × sin (θ)
Where:
• τ is the torque,
• r is the distance from the axis of rotation to the point where the force is applied (also called the lever arm),
• F is the magnitude of the force applied,
• θ is the angle between the force vector and the lever arm.
This equation represents the rotational equivalent of force in linear motion, where torque causes an object to rotate around an axis.
There is no constant of proportionality because the people creating the definition what the term "torque" should mean, decided to do it just that way.
They could have chosen an arbitrary definition and call that a "torque". But how useful is such a definition? We prefer definitions that make the description of the world easier.
In case of "torque":
This definition happened to be a good one for an "input" into rotational movements:
There are other definitions in physics where the introduction of a constant factor is useful and makes things easier, e.g. many computations related to frequency need a factor of $2 \pi$, whereas using the angular frequency gets rid of that factor. In that case, there are two competing definitions for basically the same concept, with "frequency" being more user-friendly (easier to grasp), and "angular frequency" more math-friendly (no arbitrary $2 \pi$ values appearing in formulas).
But in case of "torque", leaving out any factor was the best choice.
Basically you miss the fact that full definition of torque is this : $$ \boldsymbol{\tau} = \boldsymbol r \times \boldsymbol F $$
which is a cross-product of position vector and force vector. By definition cross-product of vectors don't need any proportionality factor or vector scaling, because it's not a simple proportionality statement, but rather a vector operation.
However, there are cases where vector scaling in cross-product is actually needed. For example, one when analyzing angular velocity sees that if we increase position vector, then tangential velocity also increases, so you can't simply state that $\boldsymbol \omega = \boldsymbol r \times \boldsymbol v$, because then $\boldsymbol \omega$ would not be a constant but rather increase too with position vector. Hence to reformulate angular velocity in terms of cross-product scaling is needed (but again, it's not a proportionality factor, just vector algebra here!) :
$$ {\boldsymbol {\omega }}={r^{-2} \cdot ( {\mathbf {r} \times \mathbf {v} }}) $$
HTH !
