Calculating compound angles manually requires specific trigonometric formulas. Here’s a breakdown of the process:

1. Determine the Number of Sides (n): This defines the basic shape of your object (e.g., 4 for a square, 6 for a hexagon).
2. Determine the Corner Angle: Calculate the angle of each corner of your shape with the formula `360 / n`. For a square, this is 90°.
3. Determine the Spring Angle: This is the angle of the slope or tilt of your workpiece relative to the wall or base. For crown molding, this is typically 38° or 45°.
4. Apply the Formulas: Use the following trigonometric formulas to find the miter and bevel settings for your saw:
   • Miter Angle = arctan(tan(Corner Angle / 2) / cos(Spring Angle))
   • Bevel Angle = arcsin(sin(Corner Angle / 2) * sin(Spring Angle))
5. Set Your Saw: Adjust your miter saw's miter and bevel settings to the calculated angles. Always perform a test cut on scrap material first.

• Compound Miter Cut: A single cut that has both a miter angle (angled across the face of the board) and a bevel angle (angled through the thickness of the board).
• Miter Angle: The angle set on the miter gauge of the saw, determining the turn of the corner.
• Bevel Angle: The angle set by tilting the saw blade, accommodating the slope of the workpiece.
• Spring Angle: The angle at which a workpiece, like crown molding, "springs" away from the surface it's mounted on. It defines the tilt.
• Hopper: A container, typically wider at the top than at the bottom, where all sides are sloped inwards. Cutting the corners of a hopper requires compound miter cuts.

"The number one rule for compound cuts is consistency. Your setup must be identical for every piece. When cutting crown molding flat on the saw, the top edge of the molding must *always* be against the fence for one side of the joint, and the bottom edge must be against the fence for the mating piece. If you mix this up, the angles won't be complementary, and your joint will never close. I always label the top and bottom edges of my molding with a pencil before I even approach the saw to avoid confusion." - Professional Trim Carpenter

Standard Crown Molding: You are installing standard 45° spring angle crown molding in a square room (4 sides).
Input: Sides = 4, Spring Angle = 45°.
The calculator will yield the classic settings for this job: a Miter Angle of 31.6° and a Bevel Angle of 33.9°.

Building a Hexagonal Pyramid: You want to build a 6-sided pyramid where the sides are tilted at a 30° angle.
Input: Sides = 6, Spring Angle = 30°.
The calculator will provide the precise miter and bevel settings needed to cut the six triangular sides so they meet perfectly at the corners.

• Incorrect Spring Angle: Using the wrong spring angle for your specific crown molding is a very common source of error and gappy joints.
• Cutting on the Wrong Side of the Line: It's easy to get confused. Always visualize which part of the board is waste before making a cut.
• Inconsistent Placement on Saw: When cutting crown molding flat, failing to keep the correct edge (top or bottom) firmly against the saw's fence will result in incorrect angles.
• Assuming Walls are Square: For crown molding, if a corner is not a perfect 90°, the standard compound angles will not work. You need to measure the corner and use a more advanced calculation or technique.

• Crown Molding Installation: The most common application, for cutting inside and outside corners.
• Custom Box Making: Creating boxes, planters, or containers with sloped sides (hoppers).
• Roof Framing: Calculating angles for hip and valley rafters in complex roof systems.
• Architectural Millwork: Fabricating custom angled structures like gazebos, pyramids, or multi-sided columns.