Calculations for Programming the Lathe 311

The class "Calculations for Programming the Lathe" provides an in-depth explanation of various calculations necessary to determine tool positions on the lathe or turning center. Trigonometry and circle geometry are used to calculate the toolpaths used in lathe cutting operations. This class introduces the foundational toolpaths and trigonometric equations, including tool nose radius compensation. It then provides a detailed explanation of the calculations needed to determine tool positions for drilling, chamfering, and turning partial and full arcs.

An understanding of trigonometry and how it can be applied on the lathe is necessary to perform any lathe operation programming. A knowledge of TNRC, drilling, and arc calculations will allow students to program most basic CNC lathe operations.

Class Details

Class Name:
Calculations for Programming the Lathe 311
Number of Lessons:
Related 1.0 Class:
Turning Calculations 285

Class Outline

  • Calculating Toolpaths and Coordinates for the Lathe
  • Trigonometric Ratios
  • Sine, Cosine, and Tangent
  • Drilling Calculations
  • Trigonometry Review
  • Tool Nose Radius Compensation
  • Chamfering Coordinates
  • Tool Nose Radius and Chamfers Review
  • Arc Programming
  • Calculating a Full Arc
  • Programming an Arc Motion
  • Arcs and Tool Nose Radius Review
  • Partial Arc Programming
  • Calculating a Partial Arc
  • Steps in Calculating Partial Arcs


  • Describe CNC toolpaths. Describe coordinates for the lathe.
  • Describe the three trigonometric ratios.
  • Describe sine, cosine, and tangent.
  • Explain how to use right triangles to calculate drilling dimensions.
  • Describe tool nose radius compensation.
  • Describe calculations for chamfering.
  • Describe programming for partial or full arcs.
  • Explain how to find the coordinates for the center of a full arc.
  • Describe the G codes required to program an arc motion.
  • Recall the calculations necessary to find the center of a partial arc.
  • Describe partial arc calculations.

Job Roles


  • CNC Turning Programming, Setup, & Operation


Vocabulary Term Definition
adjacent The side next to the reference angle in a right triangle. The adjacent side cannot be the hypotenuse.
approach A small distance that the cutting tool travels before engaging the workpiece. The approach distance is added for safety reasons.
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.
CAD/CAM Computer-aided design/computer-aided manufacturing. CAD/CAM software helps programmers to efficiently design parts and generate part programs.
Cartesian coordinate system A system of numerically locating points in three-dimensional space. CNC machines use the Cartesian coordinate system to locate the tool tip and map the dimensions of a workpiece.
centerline An imaginary line that bisects an object into two equal halves. The centerline of a drill is used to divide its tip into two equal right triangles.
chamfering Machining an angled edge around the end of a cylindrical workpiece to remove sharp corners. Chamfering generally produces a 45° angle.
chuck A device that holds a workpiece in place as it rotates on a CNC lathe. A chuck commonly has three or four jaws that can be adjusted to fit variously sized parts.
circular interpolation The toolpath required to create a circular or partially circular part. Circular interpolation necessitates simultaneous movement on at least two axes.
CNC Computer numerical control. A computerized system used to control a mill, lathe, or turning center. CNC machines are much more precise than their manual counterparts.
CNC lathe A lathe that is controlled by a computer running programs driven by numerical data. CNC lathes are much more precise than their manual counterparts.
computer numerical control CNC. A computerized system used to control a mill, lathe, or turning center. Computer numerical control machines are much more precise than their manual counterparts.
cosine The ratio of the length of the side adjacent to the angle divided by the hypotenuse. Cosine is often written as cos.
diameter The measurement of a circle or cylinder at its widest point. The diameter of a drill can be used to find the length of its tip.
drill A fluted tool designed to drill holes. Drills used for CNC lathes typically have a 118° tip.
drilling A machining operation designed to create a circular hole in a workpiece. Drilling to the correct depth requires using trigonometry to calculate the length of the drill tip.
facing An operation in which an operator uses a lathe to feed a cutting tool across an end of a cylindrical workpiece to create a flat surface. Facing is usually completed before setting program zero.
full arcs A portion of a circle that forms an angle measuring exactly 90 degrees. Cutting full arcs requires circular interpolation.
G40 The G code to turn off tool nose radius compensation. G40 should be used for any part program in which G41 or G42 has been activated, either before a toolchange or at the end of the program.
G41 The G code to turn on tool nose radius compensation. G41 compensates to the left of the toolpath.
G42 A G code to turn on tool nose radius compensation. G42 compensates to the right of the toolpath.
hypotenuse In a right triangle, the side located opposite the right angle. The hypotenuse is always the longest side.
I and K method A method for programming circular interpolation that uses an I code and a K code to indicate the coordinate locations of the arc center. The I and K method cannot be used with the R method.
linear interpolation The toolpath required to create a straight line that does not travel parallel to any axis. Linear interpolation necessitates simultaneous movement on at least two axes.
offset In CNC programming, a mathematical, spatial value that compensates for a variable that would otherwise result in an improperly sized workpiece. Offsets for the lathe include tool nose radius compensation.
origin The fixed center point of the Cartesian coordinate system. The origin has a numerical value of zero for any axis.
part program Instructions used by a CNC machine to perform the necessary sequence of operations to machine a specific workpiece. Part programs are composed of G code.
partial arcs A portion of a machined circular part. Every partial arc measures less than 90 degrees.
perpendicular An object or plane that exists at a right angle to another object or plane. Three perpendicular axes are used to define the Cartesian coordinate system.
program zero A position that acts as the origin for the part program of each particular workpiece. Program zero is selected by the part programmer.
R method A method for programming circular interpolation that uses an R code to indicate the size of the radius that forms the arc. The R method cannot be used with the I and K method.
right triangles A triangle containing one right angle, which is an angle that measures exactly 90 degrees. Right triangles are useful for calculating tool positions.
shoulder On a workpiece with different diameters, the area where it transitions from one diameter to the other. Shoulders may be curved, requiring partial arc calculations.
sine The ratio of the length of the side opposite the angle divided by the hypotenuse. Sine is often written as sin.
symmetrical An object that is identical on each side of its centerline, or for which each half is a mirror image of the other. Symmetrical parts can be machined on lathes or turning centers.
tangent The ratio of the length of the side opposite the angle divided by the adjacent side. Tangent is often written as tan.
theoretical tool tip TTT. The location from which the CNC lathe calculates the position of the tool. The theoretical tool tip may or may not be located at the actual tip of the tool.
tool nose radius TNR. Describes the rounded tip of some tools. TNR must be compensated for with TNRC.
tool nose radius compensation TNRC. An offset feature that adjusts the toolpath to accommodate the rounded tip of an insert during contouring, chamfering, and other multi-axis operations. Tool nose radius compensation may be calculated either automatically by the lathe, turning center, or CAD/CAM software or manually by the part programmer.
toolpath A series of coordinate positions that determine the movement of a tool during a machining operation. Toolpaths are expressed in G code.
trigonometric ratios Mathematical ratios from which the sides or angles of a right triangle can be calculated. Trigonometric ratios are used often in CNC lathe programming.
trigonometry The branch of mathematics that addresses the measurements and relationships of a triangle and its parts. Trigonometry is used extensively in CNC machining.
turning centers CNC lathes that are also capable of milling operations. A turning center requires calculations in all three Cartesian axes.
X axis The linear axis representing the longest distance of travel perpendicular to the spindle. The X axis typically describes forward-and-away movement on the CNC lathe or turning center.
Y axis A linear axis that is rarely used on CNC lathes. The Y axis typically describes up-and-down travel on a turning center.
Z axis The linear axis that runs parallel to the spindle, or around which the spindle rotates. The Z axis typically describes right-to-left or left-to-right travel on the CNC lathe or turning center.