By combining the polynomial transform and radix-q decomposition, the paper presents a new algorithm for the type-III r-dimensional discrete Cosine transform (rD-DCT-III) with size q l 1×q l 2× ... ×q l r , where q is an odd prime number. The number of multiplications for computing an rD-DCT-III is approximately 1/r times that needed by the row-column method while the number of additions increase slightly. The total number of operations (additions plus multiplications) is also reduced. The proposed algorithm has a simple computational structure because it needs only 1D-DCT-III and the polynomial transform.