Finding the area of polygons within other polygons in R

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Is there a function in R which will loop through a layer of zonal polygons (i.e. a polygon grid) and return the area of a second polygon layer that is contained in each grid cell? I have a layer of polygons representing areas of forest (in blue), and a second layer representing ordinance survey grid cells (in pink - each grid cell has a unique "name"). Both layers are polygon shapefiles. I would like to calculate how much forest is present in each grid cell but I cannot find a tool that will do this in R. I'm looking for something equivalent to tabulate area in ArcGIS. As you can see, the forest polygons are complex and in many cases there are several of them per grid cell, which is making it trickier.

Any advice would be much appreciated!





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