Can I use the hypotenuse of the pythagorean theorem to compute the work?
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:
$$\begin{array}{|c|c|}\text{Monoatomic}&amp;1/18\\
\text{Diatomic}&amp;1/28\\
\text{Polyatomic-Linear}&amp;1/28\\
\text{Polyatomic-Non-Linear}&amp;1/33\\
\end{array}$$