Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown.
How large a fraction does the area of the small circle make up of that of the big one?
[img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]
1. **Determine the hypotenuse of the right triangle:**
   - Given the sides of the right triangle are \(1\) and \(2\), we can use the Pythagorean theorem to find the hypotenuse \(c\):
     \[
     c = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}
     \]

2. **Find the altitude of the right triangle from the hypotenuse:**
   - The altitude \(h\) from the hypotenuse of a right triangle can be found using the formula:
     \[
     h = \frac{ab}{c}
     \]
     where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse. Substituting the given values:
     \[
     h = \frac{1 \cdot 2}{\sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}
     \]
   - This altitude is the radius of the small circle.

3. **Determine the radius of the larger circle:**
   - The radius of the larger circle is equal to the longer leg of the right triangle, which is \(2\).

4. **Calculate the ratio of the areas of the small circle to the large circle:**
   - The area of a circle is given by \(A = \pi r^2\). Let \(r_s\) be the radius of the small circle and \(r_L\) be the radius of the large circle.
     \[
     r_s = \frac{2\sqrt{5}}{5}, \quad r_L = 2
     \]
   - The area of the small circle is:
     \[
     A_s = \pi \left(\frac{2\sqrt{5}}{5}\right)^2 = \pi \left(\frac{4 \cdot 5}{25}\right) = \pi \left(\frac{20}{25}\right) = \pi \left(\frac{4}{5}\right)
     \]
   - The area of the large circle is:
     \[
     A_L = \pi (2)^2 = 4\pi
     \]
   - The ratio of the areas is:
     \[
     \frac{A_s}{A_L} = \frac{\pi \left(\frac{4}{5}\right)}{4\pi} = \frac{4}{5} \cdot \frac{1}{4} = \frac{4}{20} = \frac{1}{5}
     \]

Conclusion:
\[
\boxed{\frac{1}{5}}
\]