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This paper deals with two efficient approaches for determining the mutual inductance between thin circular coils and disk coils in air. The first approach gives new expressions for calculating the mutual inductance of treated configurations. These results are expressed over the complete elliptical integrals of the first and second kind, Heuman's Lambda function, and one term that must be solved numerically. Another approach is based on the filament method where conductors are approximated by the set of Maxwell's coils. The obtained expressions are expressed over the complete elliptical integrals of the first and second kind and permit also fast calculation of the mutual inductance for mentioned systems. These new expressions are accurate and simple for useful applications. All results obtained by the two approaches are in excellent agreement.