Calculations for Programming the Mill 312

"Calculations for Programming the Mill" provides an in-depth explanation of the various calculations necessary to program toolpaths on a CNC mill or machining center for a variety of common operations. Common CNC milling operations covered in this class are face milling, pocket milling, milling full and partial arcs, and holemaking. Important concepts for programming these toolpaths include step-over, approach distance, trigonometry, and boxing routines, as well as some of G codes.

"Calculations for Programming the Mill" details the calculations necessary to program a CNC mill. After taking this class, users will be able to understand and perform most basic CNC mill operations.

Class Details

Class Name:
Calculations for Programming the Mill 312
Number of Lessons:
Related 1.0 Class:
Milling Calculations 295

Class Outline

  • CNC Milling
  • Cartesian Coordinates and Program Zero
  • Coordinates for the Mill Review
  • Face Milling
  • Face Milling Calculations
  • Pocket Milling
  • Pocket Milling Calculations
  • Cutter Radius Compensation
  • Face Milling and Pocket Milling Review
  • Spot Drilling
  • Spot Drilling Calculations
  • Twist Drilling and Trigonometry
  • Twist Drilling Calculations
  • Trigonometry and Drilling Calculations Review
  • Bolt-Hole Patterns
  • Full-Circle Calculations
  • Final Review


  • Describe CNC milling.
  • Describe the coordinate system used for the mill. Describe program zero.
  • Describe face milling.
  • Calculate face milling cutting passes.
  • Describe pocket milling.
  • Describe calculations for pocket milling.
  • Describe cutter radius compensation.
  • Describe spot drilling.
  • Explain how to calculate drill depth for a spot drilling operation.
  • Explain how trigonometry is used in twist drilling operations.
  • Calculate total drill depth coordinate locations.
  • Explain the calculations necessary to drill a bolt-hole pattern.
  • Describe full-circle calculations.

Job Roles


  • CNC Milling Programming, Setup, & Operation


Vocabulary Term Definition
adjacent In a triangle, a side that touches an angle. The adjacent sides form the angle.
aeronautical Having to do with airplanes or other flying craft. Aeronautical applications tend to require extremely precise tolerances.
approach distance A short distance added to the beginning of a toolpath. Approach distance helps to prevent damage to the machine and the workpiece.
arc center method A method for programming circular tool movements that indicates the location of the arc's center along the X and Y axes using I and J codes. The arc center method is usually used for full-arc motions.
arc-in motion A partial-arc motion that leads into a larger-arc motion. Arc-in and arc-out motions leave a smooth surface finish.
arc-out motion A partial-arc motion that exits from a larger-arc motion. Arc-out and arc-in motions leave a smooth surface finish.
axes Imaginary lines that pass through the center of a point or object. Axes are used to describe the positions of objects on the Cartesian coordinate system.
bolt-hole pattern A series of equally spaced holes around the circumference of a larger imaginary circle. Bolt-hole patterns are calculated with trigonometry.
boxing routine A series of toolpaths used to machine a rectangular pocket. A boxing routine starts in the center of the pocket and moves outward in a rectangular pattern.
CAD/CAM Computer-aided design/computer-aided manufacturing. The use of computers to aid in the design and manufacturing of a part. CAD/CAM makes it comparatively easy to machine complex surfaces, especially parts with three-dimensional contour features.
calculator A small, handheld device used to solve mathematical problems. Calculators can be useful for finding the sine, cosine, or tangent of an angle.
Cartesian coordinate system The system that describes the position of any point or object in three-dimensional space by expressing its distance from a fixed position along linear axes. The Cartesian coordinate system is used to describe measurements in CNC milling and turning.
chamfer An angled surface added to an edge of a workpiece. Chamfers replace a sharp edge with a 45° angle.
chatter The development of surface imperfections on the workpiece caused by vibrations of the cutting tool. Chatter can occur if step-over is either too little or too much.
circumference The distance around a curve or circle. The circumference is the circle's outer perimeter.
clearance Any useful space that is intentionally maintained between components. Clearance is often necessary to prevent machine damage.
computer numerical control mills CNC mills. A milling machine that makes precise cuts according to numerical data. Computer numerical control mills are much more precise than their manual counterparts.
contour features A curved part of a workpiece. Contour features are complex and generally calculated using CAD/CAM software.
cosine In a right triangle, the ratio of the length of the side adjacent to the angle and the hypotenuse. The cosine of a right triangle can be found by calculating the ratio as a division.
cutter radius compensation CRC. An offset that accounts for variations in tool diameter. Cutter radius compensation is necessary for tools that continuously cut in the X or Y axis.
deflection The unintended movement or repositioning of a component due to a mechanical force. Deflection of a cutting tool can cause poor surface finish and inaccurate dimensions.
diameter The distance from edge to edge of the widest point of a circle. The diameter of a circle is always twice its radius.
end mill A long, thin milling cutter with a flat bottom and cutting edges that wind up the sides. End mills have helical flutes and relatively small diameters.
face mill A flat milling cutter with multiple cutting teeth on its periphery. Face mills rapidly remove metal from the top surface of a workpiece.
face milling A milling operation in which the surface of the workpiece is perpendicular to the spindle axis. Face milling is primarily used to mill the top surface of the part.
finishing pass A cutting pass that produces the surface finish and brings a feature to its proper size. The finishing pass is typically the last operation in the part program.
G code A code that determines the type of operation performed on the machine. G codes select the type of motion, speed or feed mode, etc.
geometry A branch of mathematics that involves the measurements, properties, and relationships of dimensional objects. Geometry is used in CNC machining.
hypotenuse In a right triangle, the side located opposite the right angle. The hypotenuse is the longest side of the triangle.
I code The program code that indicates the location of an arc's center along the X axis. I and J codes are used for the arc center method.
isosceles right triangle A triangle with one 90° angle and two 45° angles. The two 45° sides of an isosceles right triangle are always equal in length.
J code The program code that indicates the location of an arc's center along the Y axis. J and I codes are used for the arc center method.
length-to-diameter ratio A ratio comparing the length of a cylindrical tool to its diameter. Higher length-to-diameter ratios offer less rigidity.
machine control A computer that directs a CNC machine. The machine control interprets the part program and relays its instructions to the CNC machine.
milling cutter A multipoint tool that removes metal from the surface of a workpiece. Milling cutters can create parts with complex shapes.
offset A fixed adjustment that moves the tool to compensate for the tool's length, width, or another factor. Offsets help to ensure that finished parts are to specifications.
origin The fixed center point of the Cartesian coordinate system. The origin has a numerical value of zero for any axis.
part program A series of instructions used by a CNC machine to perform the necessary sequence of operations to machine a specific workpiece. Part programs are written in G code.
pocket An interior recess that is cut into the surface of a workpiece. Pockets can be circular, rectangular, or a combination of the two.
program zero The position that acts as the origin for the part program of each particular workpiece. Program zero is selected by the part programmer.
programming language A language used to program or provide directions to a computer or machine. The programming language used by CNC machines is called G code.
R code The program code that indicates the length of an arc's radius. In certain canned cycles, an R code indicates the R level for tool return.
radius A straight line extending from the center point to the periphery of a circle. The radius of a circle is always one-half of its diameter.
radius method A method for programming circular tool movements that indicates the size of the arc's radius using an R code. The radius method is best used for partial-arc motions.
right triangles A triangle that includes a 90° angle as one of its three angles. Right triangles are often used in drilling calculations.
sine In a right triangle, the ratio of the length of the side opposite the angle and the hypotenuse. The sine of a right triangle can be found by calculating the ratio as a division.
spot drill A short, sturdy drill used to start a hole and accurately locate it. Most spot drills have a 90° tip.
step-over The amount of the cutter's diameter that is engaged in a cut. Step-over is generally 75% or less of the cutter's diameter.
symmetrical parts A part that can be divided into two equal halves with identical features that are equal distances from the centerline. Symmetrical parts typically have program zero placed in the center.
tangent In a right triangle, the ratio of the length of the side opposite the angle and the side adjacent to the angle. The tangents of a right triangle can be found by calculating the ratio as a division.
three-dimensional Having length, width, and depth. Three-dimensional parts can be machined with CNC machines.
through holes A hole that passes completely through a workpiece. Through holes require that the depth of the hole, the length of the drill tip, and a small amount of additional clearance all be calculated.
toolpaths The series of coordinate positions that dictate the movement of a tool during a machining operation. Toolpaths are programmed using G code.
trigonometry A branch of mathematics that addresses the measurements and relationships of triangles and their components. Trigonometry is often used in milling calculations.
twist drill A common drill characterized by helical flutes along its length and two cutting edges at the drill point. Twist drilling is usually preceded by spot drilling.
X axis The linear axis representing the longest distance of travel parallel to the worktable, or left to right on a vertical mill. The X axis is perpendicular to the Y and Z axes.
Y axis The linear axis representing the shortest distance parallel to the worktable, or towards and away on a vertical mill. The Y axis is perpendicular to the X and Z axes.
Z axis The linear axis that represents motion towards and away from the worktable, or up and down on a vertical mill. The Z axis is perpendicular to the X and Y axes.