$A, B, C$, and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47$, what is the area of $ABCD$?
1. **Identify the given values and calculate the semi-perimeter:**
   - Given side lengths: \(AB = 13\), \(BC = 13\), \(CD = 37\), \(AD = 47\).
   - The semi-perimeter \(s\) of the quadrilateral is calculated as:
     \[
     s = \frac{AB + BC + CD + AD}{2} = \frac{13 + 13 + 37 + 47}{2} = \frac{110}{2} = 55
     \]

2. **Apply Brahmagupta's formula:**
   - Brahmagupta's formula for the area of a cyclic quadrilateral is:
     \[
     \text{Area} = \sqrt{(s - a)(s - b)(s - c)(s - d)}
     \]
     where \(a, b, c, d\) are the side lengths of the quadrilateral and \(s\) is the semi-perimeter.

3. **Substitute the values into the formula:**
   - Substitute \(a = 13\), \(b = 13\), \(c = 37\), \(d = 47\), and \(s = 55\):
     \[
     \text{Area} = \sqrt{(55 - 13)(55 - 13)(55 - 37)(55 - 47)}
     \]
     Simplify the terms inside the square root:
     \[
     \text{Area} = \sqrt{(42)(42)(18)(8)}
     \]

4. **Calculate the product inside the square root:**
   - First, calculate \(42 \times 42\):
     \[
     42 \times 42 = 1764
     \]
   - Then, calculate \(18 \times 8\):
     \[
     18 \times 8 = 144
     \]
   - Now, multiply the results:
     \[
     1764 \times 144 = 254016
     \]

5. **Take the square root of the product:**
   - Finally, calculate the square root of \(254016\):
     \[
     \sqrt{254016} = 504
     \]

Conclusion:
\[
\boxed{504}
\]