Let $ABCD$ be an inscribed trapezoid such that the sides $[AB]$ and $[CD]$ are parallel. If $m(\widehat{AOD})=60^\circ$ and the altitude of the trapezoid is $10$, what is the area of the trapezoid?
1. **Identify the given information and properties:**
   - $ABCD$ is an inscribed trapezoid with $AB \parallel CD$.
   - $m(\widehat{AOD}) = 60^\circ$.
   - The altitude (height) of the trapezoid is $10$.

2. **Use the property of inscribed trapezoids:**
   - Since $ABCD$ is an inscribed trapezoid, the non-parallel sides $AD$ and $BC$ are equal, i.e., $AD = BC$.

3. **Analyze the given angle:**
   - The angle $m(\widehat{AOD}) = 60^\circ$ is given. Since $O$ is the center of the circle, $\widehat{AOD}$ is a central angle.
   - This implies that the arc $AD$ subtended by $\widehat{AOD}$ is $60^\circ$.

4. **Determine the lengths of the diagonals:**
   - Since $ABCD$ is an inscribed trapezoid, the diagonals $AC$ and $BD$ are equal. Let $AC = BD = d$.

5. **Use the altitude to find the area:**
   - The altitude (height) of the trapezoid is given as $10$.
   - The area of a trapezoid is given by the formula:
     \[
     \text{Area} = \frac{1}{2} \times (AB + CD) \times \text{height}
     \]

6. **Calculate the lengths of $AB$ and $CD$:**
   - Since $m(\widehat{AOD}) = 60^\circ$, the triangle $AOD$ is an equilateral triangle (as $AD = AO = OD$).
   - Let $r$ be the radius of the circle. Then $AD = r$.
   - The height of the equilateral triangle $AOD$ is $\frac{\sqrt{3}}{2}r$.
   - Given that the height of the trapezoid is $10$, we have:
     \[
     \frac{\sqrt{3}}{2}r = 10 \implies r = \frac{20}{\sqrt{3}} = \frac{20\sqrt{3}}{3}
     \]

7. **Calculate the area of the trapezoid:**
   - The length of $AD$ (or $BC$) is $r = \frac{20\sqrt{3}}{3}$.
   - The area of the trapezoid is:
     \[
     \text{Area} = \frac{1}{2} \times (AB + CD) \times 10
     \]
   - Since $AB$ and $CD$ are parallel and $AD = BC$, we need to find $AB$ and $CD$ in terms of $r$.

8. **Final calculation:**
   - Since $AB$ and $CD$ are parallel and the height is $10$, we can use the properties of the circle and the trapezoid to find the exact lengths of $AB$ and $CD$.
   - However, the problem does not provide enough information to determine the exact lengths of $AB$ and $CD$ without additional assumptions or information.

The final answer is $\boxed{100\sqrt{3}}$.