1. **Stating the problem:** We have a square with side length $R=20$ cm. Inside the square, a diagonal arc runs from the upper-left corner to the lower-right corner, and the angle formed at the lower-left corner is $30^\circ$. We want to analyze this geometric configuration. 2. **Understanding the problem:** The square has all sides equal to $20$ cm. The diagonal of the square connects opposite corners, and the angle at the lower-left corner is marked as $30^\circ$. This suggests we are dealing with a right triangle formed by the diagonal and the sides of the square. 3. **Formula and rules:** The diagonal $d$ of a square with side length $R$ is given by: $$d = R\sqrt{2}$$ The angle between the diagonal and the base in a square is normally $45^\circ$, but here it is $30^\circ$, indicating a different geometric setup or an arc inside the square. 4. **Calculations:** Given the angle $30^\circ$ at the lower-left corner, we can analyze the triangle formed by the base $R=20$ cm, the diagonal arc, and the vertical side. Using trigonometry, if the angle at the lower-left corner is $30^\circ$, then: $$\tan(30^\circ) = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\text{vertical side}}{20}$$ Since $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, we have: $$\frac{1}{\sqrt{3}} = \frac{\text{vertical side}}{20}$$ Solving for the vertical side: $$\text{vertical side} = \frac{20}{\sqrt{3}} = \frac{20\sqrt{3}}{3} \approx 11.55 \text{ cm}$$ 5. **Summary:** The vertical side corresponding to the $30^\circ$ angle is approximately $11.55$ cm, while the base remains $20$ cm. 6. **Final answer:** The vertical side length corresponding to the $30^\circ$ angle in the square with base $20$ cm is: $$\boxed{\frac{20\sqrt{3}}{3} \approx 11.55 \text{ cm}}$$