We are given some similar triangles.  Their areas are $1^2,3^2,5^2,\cdots,$ and $49^2$.  If the smallest triangle has a perimeter of $4$, what is the sum of all the triangles' perimeters?
1. **Identify the given areas of the triangles:**
   The areas of the triangles are given as \(1^2, 3^2, 5^2, \ldots, 49^2\). This is an arithmetic sequence of squares of odd numbers from \(1\) to \(49\).

2. **Determine the ratio of the perimeters:**
   Since the triangles are similar, the ratio of the perimeters of the triangles is the same as the ratio of the square roots of their areas. For the smallest triangle, the area is \(1^2 = 1\) and the perimeter is given as \(4\).

3. **Calculate the ratio of perimeter to the square root of the area:**
   For the smallest triangle:
   \[
   \text{Perimeter} = 4, \quad \text{Area} = 1^2 = 1
   \]
   The square root of the area is:
   \[
   \sqrt{1} = 1
   \]
   Therefore, the ratio of the perimeter to the square root of the area is:
   \[
   \frac{4}{1} = 4
   \]

4. **Apply the ratio to find the perimeters of all triangles:**
   For each triangle with area \(n^2\), the perimeter \(P\) is:
   \[
   P = 4 \times n
   \]
   The areas given are \(1^2, 3^2, 5^2, \ldots, 49^2\). Thus, the corresponding perimeters are:
   \[
   4 \times 1, 4 \times 3, 4 \times 5, \ldots, 4 \times 49
   \]
   This forms the sequence:
   \[
   4, 12, 20, \ldots, 196
   \]

5. **Sum the perimeters:**
   The sequence of perimeters is an arithmetic sequence with the first term \(a = 4\) and the common difference \(d = 8\). The number of terms \(n\) can be found by noting that the last term is \(196\):
   \[
   4 + (n-1) \cdot 8 = 196
   \]
   Solving for \(n\):
   \[
   4 + 8n - 8 = 196 \implies 8n - 4 = 196 \implies 8n = 200 \implies n = 25
   \]

6. **Sum the arithmetic sequence:**
   The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by:
   \[
   S_n = \frac{n}{2} \left(2a + (n-1)d\right)
   \]
   Substituting the values:
   \[
   S_{25} = \frac{25}{2} \left(2 \cdot 4 + (25-1) \cdot 8\right) = \frac{25}{2} \left(8 + 192\right) = \frac{25}{2} \cdot 200 = 25 \cdot 100 = 2500
   \]

The final answer is \(\boxed{2500}\).