A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$?

[i]Proposed by Dylan Toh[/i]
1. **Understanding the Problem:**
   We need to find the expected value of the area of the triangle $T$ formed by the ends of the hour, minute, and second hands of a clock. Each hand has a length of $1$.

2. **Reformulating the Problem:**
   The problem can be reformulated as finding the expected area of a triangle whose vertices are uniformly chosen on the unit circle. This is because the ends of the clock hands can be considered as points on the unit circle.

3. **Using Known Results:**
   According to the result from the reference provided (see [here](https://mathworld.wolfram.com/CircleTrianglePicking.html)), the expected area of a triangle formed by three points chosen uniformly at random on the unit circle is given by:
   \[
   \mathbb{E}([T]) = \frac{3}{2\pi}
   \]

4. **Conclusion:**
   Therefore, the expected value of the area of the triangle $T$ formed by the ends of the hour, minute, and second hands of the clock is:
   \[
   \mathbb{E}([T]) = \frac{3}{2\pi}
   \]

The final answer is \(\boxed{\frac{3}{2\pi}}\).