The nearest neighbors (NN) algorithm is a traditional algorithm utilized in many research areas like computer graphics, classification and/or machine learning. This paper aims at the fixed-radius nearest neighbors algorithm in 2D space. The neighbors are searched within a circular neighborhood which is positioned in the bounded space. The radius of the circle is known in advance. The uniform grids can be efficiently utilized for a nearest points query acceleration. This paper presents a study comparing the square and the hexagonal uniform grids and their suitability for a circular neighborhood querying. The two algorithms checking the mutual position/intersection of a circle and a square or a hexagon are described. The tests show the supremacy of the hexagonal grid.