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Like any surface, it's easiest to wipe up spills as you go. If you wait to wipe up the spill, it can dry on the surface, making it harder to clean. Since Corian countertops are nonporous, the spill won't soak in, making it a breeze to just wipe it up with a clean, damp cloth. Always dry the surface immediately after cleaning. If you don't dry the surface, film will build up over time. If the spill has dried, you can use warm, soapy water to clean it up. Put a bit of dish soap on a rag, then add warm water to make it sudsy. Wipe it over the spill or residue to clean it up, then rinse it with clean warm water. If regular soap and water don't remove the stain, you can use an ammonia-based cleaner. Spray or rub the cleaner in, then wipe it up. Make sure to rinse it with clean water, and don't forget to dry it so you don't leave streaks.  Don't use window cleaners on this type of counter, as these can leave a waxy buildup. Before using any cleaner, read the directions on the bottle. Always test the cleaner first on an out-of-way corner, so you can rest assured it won't create a different problem. Although glass cleaner does have ammonia in it, you should skip it for this purpose. It can leave a film on the countertop. Pick a household cleaner instead. If you don't have an ammonia-based cleaner, you can use another non-abrasive cleaner. It must be non-abrasive because Corian countertops are easily scratched. Spray the cleaner on the counter, and use a clean, wet rag to rub the cleaner in.  You can leave the cleaner on for a minute or two to help remove the stain. Rinse and dry the countertop when you're done. Make sure not to mix different types of cleaners, as that can create toxic fumes.
Wipe up spills when they happen. Use warm water and a dishcloth or sponge. Try an ammonia-based cleaner. Pick another non-abrasive cleaner.