How can we estimate the result size for a given query on complex spatial objects like polygons? Estimating a query's result size, also known as the cardinality estimation, plays a significant role in query scheduling and optimization. Accurate and fast cardinality estimation substantially improves query efficiency. Existing compatible solutions, mainly histogram-based, deal with polygons as their minimal bounding rectangles for easier processing, which leads to inaccurate estimation. To address this issue, we present PolyCard, a learned cardinality estimator for intersection queries on spatial polygons. We successfully apply learning techniques to spatial polygons with variable sizes. PolyCard has the following properties. (i) Accurate: PolyCard improves 30% accuracy compared with existing solutions, (ii) Fast: PolyCard takes only 4 microseconds for an estimation, and (iii) Stable: PolyCard is robust against datasets and queries of different cardinalities. Our experiments on four real-world datasets of millions of polygons demonstrate the efficiency and effectiveness of PolyCard.