# Compound Angle Calculator

A table saw or compound miter saw can cut workpieces with two angle settings; bevel and miter. Such a saw is useful when building for example boxes with slanted sides or concrete forms for post caps. It is surprisingly complex to compute compound angle settings. On this page I have collected a few compound angle calculators that will help compute compound angles.

### Definitions:

• A compound cut consists of two angles, the bevel angle and the miter angle.
• The bevel angle (or blade tilt) is the tilt of the saw blade from vertical on the saw table. This means that a normal square cut has a bevel of 0°. Typically saws have a maximum bevel of 45°.
• The miter angle (or cross-cut angle) is the horizontal angle, as seen on the saw table, from a line perpendicular to the long edge of a board. The miter angle is set on the miter gauge of the table saw. A perpendicular cut has a miter of 0°.
• Some saws label the miter angle differently, with a perpendicular cut labeled 90°. This is the miter complement angle.
• The dihedral angle is the angle between two surface planes. Specifically, we are computing the inside dihedral angle, which is always ≤180°. A block that fits snugly between the two surfaces can be cut with the miter angle set to the dihedral angle minus 90° and with zero blade tilt.
• A miter joint joins the cut ends of two boards.
• A butt joint joins the cut end of one board (butt) with the uncut side of a second board (cap).
• A negative angle shows that this angle is in the opposite direction of a positive angle, for example the blade tilts right instead of left. This occurs mainly in butted joints because the blade typically tilts in the opposite direction from mitered joints.

Miter joints and butt joints have identical miter angles; only the bevel angle is different. We can see this in the picture on the right; the cutting lines on the face of the board are parallel.

Angle Precision: Choose the number of decimal digits you would like to round the angle results to.

 Number of Decimals in Angles 0      1      2      3

## Post and Board

A board with a slope in the long direction intersects a vertical post which is rotated a certain angle.

 Slope angle: Post rotation angle: Board rotation angle:

 Blade tilt: Miter angle: Miter complement: Dihedral angle:

If your slope or rotation goes in the opposite direction than shown then use a negative angle.

If the board rotation is 0° then the cut can be done without math. Set the blade tilt to the post rotation angle and the miter gauge to the slope angle, then put the board on edge (roll the board 90°) on the saw.

## N-sided Box

An n-sided box is built from n identical side pieces and a bottom. The box can have sides that are slanted outwards. The outward angle is the side angle.

 Number of sides: Side angle (deg): from: vertical horizontal

 Blade tilt, mitered joint: Blade tilt, butted joint: Miter angle: Miter complement: Dihedral angle:

Note that this calculator also works for rectangular boxes.
Derivation

## N-sided Pyramid

An n-sided pyramid is built from n identical triangular side pieces, not including the base. The base radius is the distance from the center of the pyramid's base to one of the base corners. The height is the distance from the center of the base to the apex.

 Number of sides: Base radius: Pyramid height:

 Blade tilt, mitered joint: Blade tilt, butted joint: Miter angle: Miter complement: Base side: Side Slope: Dihedral angle:

The side slope is measured from horizontal and can be used to calculate any cuts needed at the bottom of the pyramid's sides.

Pyramids with butted joints are some weird animals. A few more or less realistic examples are shown on the right.

## Rectangular Pyramid

A rectangular pyramid is a pyramid with a rectangular base.

A concrete form for casting post caps can be made in the shape of a pyramid. The pyramid can have a square or rectangular base.

 Base side X: Base side Y: Pyramid height:

Side: X Y
Blade tilt, butted joint: ← Same
Miter angle:
Miter complement:
Side slope:
Dihedral angle: ← Same

The side slopes are measured from horizontal and can be used to calculate any cuts needed at the base of the pyramid.

In the case of a concrete form, we need the inside surfaces of the form to line up. In other cases you may want the outside surfaces to line up, but given the same board thickness and a rectangular base, we cannot have both the inside and the outside surfaces line up with this model. For that special case, check out the Rectangular Frustum model below.

## Rectangular Frustum

In geometry, a frustum is similar to a prism, but the sides are not parallel. This example has a base and a top with different aspect ratios.

Side: X Y
Base:
Top:
Height:

Side: X Y
Blade tilt, mitered joint: ← Same
Blade tilt, butted joint: ← Same
Miter angle:
Miter complement:
Side slope:
Dihedral angle: ← Same

The side slopes are measured from horizontal and can be used to calculate any cuts needed at the base and top of the frustum.

Both the inside and outside surfaces line up in this model, on account of the bevel angle being the same for all pieces. This works well as long as there is an opening at the top. Once that opening closes, we must make additional cuts to clear interference between the pieces. The mitered joint Rectangular Pyramid model does not have this limitation. If you want to try this model with a pyramid, set both top dimensions to zero.

## Platonic Solids

The platonic solids is a group of five polyhedra constructed with identical sides.

 Sides 4 6 8 12 20 Name Tetrahedron Cube Octahedron Dodecahedron Icosahedron Side shape Equilateral triangle Square Equilateral triangle Pentagon Equilateral triangle Blade tilt, mitered joint: 54.736° 45° 35.264° 31.717° 20.905° Miter angle: 30° 0° 30° 18° 30° Miter complement: 60° 90° 60° 72° 60° Dihedral angle: 70.529° 90° 109.471° 116.565° 138.190°

## General Compound Angles

In the general case we have two intersecting surface planes. By defining four points, 0 to 3, on the surfaces in 3D space we can find the miter and bevel angles of sides A and B.

The figure on the right shows the two pieces assembled. The dashed lines show how the XYZ coordinates for point 1 are determined.

Points 0 and 1 lie on the seam between the two surfaces. Point 1 can be located anywhere along that seam except at the origin (point 0).

The line between points 0 and 2 is the reference edge for surface A's miter angle. Point 2 can be located anywhere along that edge except at the origin. Line 0-3 likewise for surface B.

X Y Z
Point 0: 0 0 0
Point 1:
Point 2:
Point 3:

Side: A B
Blade tilt, mitered joint: ← Same
Blade tilt, butted joint: ← Same
Miter angle:
Miter complement:
Dihedral angle:

For the mitered joint, for simplicity the blade tilts for pieces A and B are the same, but you can change them as long as their sum is the same. This will affect the intersection of A and B at the top and bottom. For the butted joint the blade tilts must be the same.

## General Compound Angles 2

Two boards have been joined at arbitrary angles. We orient the joined boards such that their edges are parallel to the horizontal plane. We now specify three angles:

 Angle between the boards in the horizontal plane (deg): Side angle of board A with respect to horizontal (deg): Side angle of board B with respect to horizontal (deg):

Side: A B
Blade tilt, mitered joint: ← Same
Blade tilt, butted joint: ← Same
Miter angle:
Miter complement:
Dihedral angle:

## Flip Board On Edge

The calculators on this page assume that the boards are laying flat on the saw table. Sometimes we would like to have the board lay with its narrow side down instead. Specifically, we flip the board so that the surface that was facing away from us is now facing up. See the example on the right for how the signs of the angles are defined.

To use this calculator, either first use another calculator above and the inputs will be filled in automatically, or enter values in the input boxes manually. For the simplest case, just one blade tilt and a miter angle are needed.

Board Flat
Board: A B
Miter angle (deg):

Board On Edge
Board: A B
Miter angle:
Miter complement:
Miter angle:
Miter complement:

The following methods may be useful for orienting the board for the compound angles.

• A (-bevel, -miter): Flipping the board 180° (rolling the board on its axis), both blade tilt and miter angle change sign. This is useful for example if the saw can only tilt the blade in one direction.
• B (-miter): Flipping the board end-for-end changes sign of only the miter angle, but it also moves the waste piece to the other side of the blade. This may or may not be practical depending on the situation.
• C (-bevel): Rotating the board 180° horizontally (same side up) changes the sign of only the blade tilt, and it moves the waste piece to the other side of the blade.

## Effective Kerf

Kerf is the width of the slot cut by the saw blade. When we make a compound cut the width of the slot increases on the surfaces of the board. Bevel affects the width of the top surface slot, while both bevel and miter affect the width of the slot in the longitudinal direction of the board.

If we want to lay out several pieces to be cut on a board, then we need to know the effective longitudinal kerf which will add up considerably over a few workpieces.

 Saw kerf: Miter (deg): Bevel (deg):

 Effective top kerf: Effective longitudinal kerf: