Can I use the hypotenuse of the pythagorean theorem to compute the work? A friend of mine told me that she used the hypotenuse of the pythagorean theorem to compute the work in graph below and the to compute the coefficient of efficiecy in this thermodynamics triangle cycle. What do you tnink about?
The work would be area of triangle, which is no way related to the hypotenuse (other than $\small H^2=P_0^2+V_0^2$)$$W=\frac12P_0V_0$$
Heat is :
$$Q_{in:A-C}=(C_V+R/2)\left(3\frac{P_0V_0}{R}\right)+C_V\left(2\frac{P_0V_0}{R}\right)$$
Efficiency is :
$$\eta=\frac{W}{Q_1}=\frac{\frac12P_0V_0}{(C_V+R/2)\left(3\frac{P_0V_0}{nR}\right)+C_V\left(2\frac{P_0V_0}{R}\right)}=\frac12.\frac{R}{3C_V+3R/2+2C_V}$$
$$\eta=\frac{R}{10C_V+3R}=\frac{R}{5fR+3R}=\frac1{5f+3}$$
which for  a monoatomic,diatomic and polyatomic(linear/non-linear) are:
$$\begin{array}{|c|c|}\text{Monoatomic}&amp;1/18\\
\text{Diatomic}&amp;1/28\\
\text{Polyatomic-Linear}&amp;1/28\\
\text{Polyatomic-Non-Linear}&amp;1/33\\
\end{array}$$