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Yes, Corian® and other solid surface countertops can be cut with standard woodworking tools like a circular saw or a jigsaw, but using the correct blade is crucial for the best results.

The recommended blade is a tungsten carbide-tipped blade, often called a solid-surface planer or triple chip grind blade. This type of blade is hard enough to cut through the material effectively. For a jigsaw, a fine-tooth down-cutting blade should be used.

It is important to note that a diamond blade is not necessary. Diamond blades are designed for harder materials like natural stone, concrete, or brick, and Corian is not a natural stone.

Using blades designed for wood is not recommended as they can cause microfractures that weaken the countertop's edges. While a tungsten carbide blade provides a much cleaner cut, the resulting edge will still be rough and must be sanded and smoothed to prevent microfractures from developing over time.

If you don’t have a power saw with a blade where you can adjust the angle, you could use a miter box and hand saw to make the cuts. The miter box contains wide slots to support the piece of baseboard and other narrow slots to accommodate the blade of the hand saw. However, it’s tough to make non-standard cuts beyond 30 or 45 degrees when using a miter box.

Yes, you can find ceiling joists using simple tools if a stud finder isn't available.

Using a hammer requires patience and a keen ear. Gently tap the ceiling in a grid pattern, listening for sound changes. A hollow sound indicates space between joists, while a solid, dull thud suggests a joist. Mark the location with a pencil and confirm by tapping from different angles.

Alternatively, you can use a nail, but this method will leave small holes. You'll feel increased resistance when the nail hits a joist. Mark the spot and repeat the process every 16 or 24 inches to map the joist locations. This method is considered less reliable than using a stud finder.

There are two ways you can cut trim for doorways: with a miter saw or a table saw. Mitered joints need a 45-degree angle. If you’re unfamiliar with a miter saw, you might want to consider buying extra trim pieces or choosing butted joints instead. One wrong cut, and you’ll need to power up the saw again with a new piece. 

Butted joints—also called shaker style—don’t need a 45-degree angle. Instead, your side pieces attach perpendicularly to the header trim and only require a straight cut.