Inferring the drivable area in a scene is crucial for ensuring a vehicle avoids obstacles and facilitates safe autonomous driving. In this paper, we concentrate on detecting the instantaneous free space surrounding the ego vehicle, targeting short-range automotive applications. We introduce a novel polygon-based occupancy representation, where the interior signifies free space, and the exterior represents undrivable areas for the ego-vehicle. The radar polygon consists of vertices selected from point cloud measurements provided by radars, with each vertex incorporating Doppler velocity information from automotive radars. This information indicates the movement of the vertex along the radial direction. This characteristic allows for the prediction of the shape of future radar polygons, leading to its designation as a ``deformable radar polygon". We propose two approaches to leverage noisy radar measurements for producing accurate and smooth radar polygons. The first approach is a basic radar polygon formation algorithm, which independently selects polygon vertices for each frame, using SNR-based evidence for vertex fitness verification. The second approach is the radar polygon update algorithm, which employs a probabilistic and tracking-based mechanism to update the radar polygon over time, further enhancing accuracy and smoothness. To accommodate the unique radar polygon format, we also designed a collision detection method for short-range applications. Through extensive experiments and analysis on both a self-collected dataset and the open-source RadarScenes dataset, we demonstrate that our radar polygon algorithms achieve significantly higher IoU-gt and IoU-smooth values compared to other occupancy detection baselines, highlighting their accuracy and smoothness.