Slenderness Ratio:
The slenderness ratio of a section is a dimensionless parameter that quantifies the relative slenderness of a structural member. It is defined as the ratio of the effective length of the member to its radius of gyration. The slenderness ratio is an important factor in determining the stability and buckling behavior of structural elements.

Explanation:
To understand why the slenderness ratio is inversely proportional to the radius of gyration, let's consider a simple example. Imagine a slender column and a short column made of the same material with the same cross-sectional area.

- Slender Column: The slender column will have a larger slenderness ratio because its effective length is much greater compared to its radius of gyration. This means that the column is relatively long and slender.

- Short Column: The short column will have a smaller slenderness ratio because its effective length is much smaller compared to its radius of gyration. This means that the column is relatively short and stout.

Effect of Slenderness on Buckling:
When a structural member is subjected to compressive loads, it may undergo buckling, which is a sudden lateral deflection or failure due to instability. The slenderness ratio plays a crucial role in determining the critical buckling load of a member.

- Slender Column: A slender column with a high slenderness ratio is more prone to buckling because it has a lower critical buckling load. The slender column is less able to resist compressive forces and is more likely to fail due to buckling.

- Short Column: A short column with a low slenderness ratio is less prone to buckling because it has a higher critical buckling load. The short column is more able to resist compressive forces and is less likely to fail due to buckling.

Relationship with Radius of Gyration:
The radius of gyration is a measure of how the area of a section is distributed about an axis. It is related to the distribution of material away from the centroid of the section. The greater the radius of gyration, the more the material is distributed away from the centroid.

- Inverse Proportion: The slenderness ratio is inversely proportional to the radius of gyration. This means that as the radius of gyration increases, the slenderness ratio decreases, and vice versa.

- Intuition: Intuitively, a larger radius of gyration implies that the material is distributed further away from the centroid, making the section more resistant to buckling. Therefore, a larger radius of gyration corresponds to a lower slenderness ratio, indicating a shorter and stouter member that is less prone to buckling.

Conclusion:
In conclusion, the slenderness ratio of a section is inversely proportional to the radius of gyration. The slenderness ratio quantifies the relative slenderness of a member, and a higher slenderness ratio indicates a greater likelihood of buckling. The radius of gyration measures the distribution of material away from the centroid, and a larger radius of gyration corresponds to a lower slenderness ratio, indicating a section that is less prone to buckling.