<p>Circular intuitionistic fuzzy sets (CIFSs) are recent extension of intuitionistic fuzzy sets (IFSs), introduced by Atanassov in 2020. In CIFS, every element is specified by a circle, so in addition to membership and non-membership degrees, one more factor, which is the radius of the circle, is also added, which helps to handle the uncertain information in a better way. Many distance measures have been defined for IFSs; only a few exist for CIFSs. Existing measures have certain limitations, like uniform radius across all elements or high computational complexity. Due to the uniform radius, these measures are not applied to CIFSs with different radii. To address these limitations, new measures are introduced by extending the cosine function-based intuitionistic fuzzy distance measure. These new approaches are based on the three term distance measure, which considers membership degree, non-membership degree and radius and the four term distance measure, which incorporates the membership degree, non-membership degree, hesitation degree and radius. New measures satisfy the axioms of a distance metric and are supported by various theorems. These measures are applied to a decision-making problem related to selecting a suitable site for an epidemic hospital. Robustness and sensitivity analyses are conducted to check the effectiveness of the proposed measures. Finally, comparisons are made with the existing distance measures and approaches to demonstrate the effectiveness of the proposed measures. The proposed measures can be applied to various problems related to pattern recognition, medical diagnosis and multi-criteria decision making (MCDM). However, these measures slightly increase the complexity of the system.</p>