I can't set the width and height of a polygon
How to do that in QGIS?
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An alternative to Willy's excellent suggestion of the "Improved Polygon Capturing" plugin, depending on your circumstances, is the "Rectangles, Ovals Digitizing" tool.
There is yet another plugin to consider (there's always more than one way to skin a cat in GIS!). This is the similarly named "Rectangles, Ovals, Diamonds" tool. This tool is actually a buffering tool rather than a digitizing one. If you have a lot of polygons to produce where you already know the centroids of each polygon, then this tool can prove very effective. You create a point data set and add width, height and rotation fields. The tool then creates a polygon data set by buffering the points according to the values in your three new attribute columns. However, the buffers this tool creates are not circular but square or elliptical (diamonds being obviously merely rotated squares).