BIZOMAT MAKEOS · Geometry
Polar and Rectangular Coordinates
What are the X–Y coordinates of this radius and angle, or the radius and angle of this X–Y point?
Convert a point between polar (radius and angle) and rectangular (X and Y) coordinates about a chosen origin, for hole positions and CNC programming.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Convert | choice | Direction of conversion. |
| Radius | mm | Distance from the origin (used for radius and angle → X and Y). |
| Angle from +X axis (anticlockwise) | ° | Angle measured anticlockwise from the positive X axis; negative is clockwise. |
| X | mm | X coordinate from the origin (used for X and Y → radius and angle). |
| Y | mm | Y coordinate from the origin. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Convert
- What it is
- The direction of the conversion: from a radius and angle to X and Y positions, or from X and Y positions to a radius and angle.
- Units and format
- Choose one of 2 options from the list: Radius and angle → X and Y; X and Y → radius and angle.
- Example
- The form opens with “Radius and angle → X and Y” selected as a worked example. Choose the option that matches your case.
- Get it right
- Only the inputs for the chosen direction are used.
What this means: Radius
- What it is
- The straight distance from the origin (for example a bolt circle centre) to the point. Used when converting radius and angle to X and Y.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For a bolt circle this is half the bolt circle diameter.
What this means: Angle from +X axis (anticlockwise)
- What it is
- The direction of the point from the origin, measured anticlockwise from the positive X axis. Used when converting radius and angle to X and Y.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -36000° to 36000°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Check where the drawing puts zero and which way it measures; a clockwise angle is entered as a negative number.
What this means: X
- What it is
- The point’s horizontal position from the origin along the X axis, positive to the right. Used when converting X and Y to radius and angle.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 43.3013 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Keep the sign: a point left of the origin has a negative X, which changes the angle.
What this means: Y
- What it is
- The point’s vertical position from the origin along the Y axis, positive upwards. Used when converting X and Y to radius and angle.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Keep the sign: a point below the origin has a negative Y.
Method and formulas
- X = r cos θ; Y = r sin θ.
- r = √(X² + Y²); θ = atan2(Y, X), reported from 0° to 360°.
- Example: a radius of 50 mm at 30° is X = 43.301 mm, Y = 25 mm.
Limitations
- Angles follow the mathematical convention (anticlockwise from +X); some machine controls and drawings measure differently.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Program a hole at a radius and angle
- Check a measured XY position against a polar callout
- Convert inspection data
Who it is for
- CNC programmers
- Quality inspectors
- Toolmakers
What you get
- X and Y
- Radius
- Angle (0–360°)
Interpreting the result
- Confirm the angle zero and direction against the drawing before programming.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 43–45, 692–694. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Absolute, relative and polar points
In Drafting Geometry by the Numbers
What it covers: Enter points by absolute, relative and polar coordinates, convert between them exactly, and avoid the angle, sign and quadrant errors that put features on the wrong side of a part.
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