A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$. Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular.
1. **Define the vertices of the pentagon**: Let the vertices of the pentagon \(P = ABCDE\) be inscribed in a circle of radius 1. We can place the vertices on the unit circle in the complex plane as follows:
   \[
   A = (1, 0), \quad B = (\cos \beta, \sin \beta), \quad C = (\cos \gamma, \sin \gamma), \quad D = (\cos \delta, \sin \delta), \quad E = (\cos \epsilon, \sin \epsilon)
   \]

2. **Determine the slopes of the chords**: The slope of the chord \(AC\) is given by:
   \[
   m_{AC} = \frac{\sin \gamma - 0}{\cos \gamma - 1} = -\cot \frac{\gamma}{2}
   \]
   Similarly, the slope of the chord \(BD\) is:
   \[
   m_{BD} = \frac{\sin \delta - \sin \beta}{\cos \delta - \cos \beta} = -\cot \frac{\beta + \delta}{2}
   \]

3. **Condition for perpendicular chords**: Since \(AC \perp BD\), we have:
   \[
   \beta + \delta = \pi + \gamma \quad (1)
   \]

4. **Area of the pentagon**: The area of the pentagon \(ABCDE\) can be calculated using the formula for the area of a polygon inscribed in a circle:
   \[
   \text{Area} = \frac{1}{2} \left[ \sin \beta + \sin (\gamma - \beta) + \sin (\delta - \gamma) + \sin (\epsilon - \delta) - \sin \epsilon \right]
   \]

5. **Simplify the area expression using (1)**: Using the condition \(\beta + \delta = \pi + \gamma\), we can rewrite the area as:
   \[
   \text{Area} = \frac{1}{2} \left[ 2 \sin \beta - \sin \delta + \sin (\epsilon - \delta) - \sin \epsilon \right] = f(\beta, \delta, \epsilon)
   \]

6. **Find the critical points**: To maximize the area, we take partial derivatives and set them to zero:
   \[
   \frac{\partial f}{\partial \beta} = \cos \beta = 0 \Rightarrow \beta = \frac{\pi}{2}
   \]
   \[
   \frac{\partial f}{\partial \delta} = -\cos \delta - \cos (\epsilon - \delta) = 0 \Rightarrow 2 \delta = \pi + \epsilon
   \]
   \[
   \frac{\partial f}{\partial \epsilon} = \cos (\epsilon - \delta) - \cos \epsilon = 0 \Rightarrow 2 \epsilon = 2 \pi + \delta
   \]

7. **Solve the system of equations**: Solving the system of equations:
   \[
   \delta = \frac{4\pi}{3}, \quad \epsilon = \frac{5\pi}{3}, \quad \gamma = \frac{5\pi}{6}
   \]

8. **Calculate the maximum area**: Substituting these values back into the area formula, we get:
   \[
   \text{Maximum Area} = 1 + \frac{3\sqrt{3}}{4}
   \]

The final answer is \(\boxed{1 + \frac{3\sqrt{3}}{4}}\).