1. **State the problem:** We are given two equations: $$A = \frac{\pi}{4} d^2$$ $$\tau = \frac{P}{A}$$ and we want to derive an expression for $\tau$ in terms of $P$ and $d$. 2. **Understand the formulas:** - $A$ is the cross-sectional area of a circular object with diameter $d$. - $\tau$ is the shear stress, defined as force $P$ divided by area $A$. 3. **Substitute $A$ into the equation for $\tau$:** $$\tau = \frac{P}{A} = \frac{P}{\frac{\pi}{4} d^2}$$ 4. **Simplify the expression:** $$\tau = \frac{P}{\frac{\pi}{4} d^2} = \frac{P}{1} \times \frac{4}{\pi d^2} = \frac{4P}{\pi d^2}$$ 5. **Final derived formula:** $$\boxed{\tau = \frac{4P}{\pi d^2}}$$ This formula expresses the shear stress $\tau$ in terms of the applied force $P$ and the diameter $d$ of the circular cross-section. This completes the derivation.