Equal-scale coordinate plane with ordered vertices and bounding boxUpdates with every input
Math & Statistics
Polygon Graphing Calculator
Generate the ordered Cartesian vertices of a translated and rotated regular polygon, inspect its bounding box and centroid, and graph the actual coordinate geometry on equal axes.
Side length -
Perimeter -
Shoelace area -
Vertex centroid -
Bounding width -
Bounding height -
Traversal orientation -
Closure residual -
CARTESIAN VERTEX MAP
Equal-scale coordinate plane with ordered vertices and bounding box
Translation, rotation, and radius redraw the polygon without distorting the axes; vertex labels follow traversal order.
VERTEX REGISTER
Coordinates, polar angles, and outgoing edge vectors
The ordered list is suitable for plotting, CAD preparation, or checking a coordinate-based implementation.
| Vertex | Angle | x | y | Next-edge vector | Edge length |
|---|
PLOTTING SETUP
Choose the coordinate frame before copying vertices
- Enter the whole vertex count.
- Use circumradius rather than apothem.
- Set the intended center coordinates.
- Rotate the first vertex to match the drawing convention.
- Copy vertices in their displayed counterclockwise order.
COORDINATE INVARIANTS
Translation and rotation move the drawing but preserve its intrinsic geometry
Changing the center adds the same offset to every vertex. Changing rotation adds the same angular offset. Neither operation changes side length, perimeter, or area.
The bounding box can change with rotation even when the polygon itself does not. That distinction matters for layout and packaging.
Subject fundamentals
Five ideas that control this calculation
Vertices follow one angular sequence
Each vertex advances by 2pi/n radians from the entered rotation, so the coordinate order remains explicit and reproducible.
Translation changes position only
Adding the center coordinates to every vertex moves the polygon without changing its side length, perimeter, area, or orientation.
Rotation changes the envelope
The polygon remains congruent while its axis-aligned width and height can change, which matters when fitting it into a rectangular region.
Shoelace area depends on order
The signed cross-sum assumes consecutive vertices. Sorting the coordinates independently would destroy the boundary path and invalidate the area.
Closure is a coordinate audit
The final edge must return from the last vertex to the first, and the vertex centroid should recover the entered center within floating-point precision.
POLAR-TO-CARTESIAN TRANSFORM
Rotate on the circle, then translate into the requested coordinate frame
Every vertex uses the same radius and an angle increment of 2pi/n. The shoelace sum independently recovers the polygon area from the generated coordinates.
Detailed calculation process and general formulas
theta_i = phi + 2pi i / nx_i = c_x + R cos(theta_i)y_i = c_y + R sin(theta_i)s = 2R sin(pi/n)K = |sum(x_i y_{i+1} - y_i x_{i+1})| / 2Symbols, meanings, and units
- i
- zero-based vertex indexcount
- phi
- entered first-vertex rotationradians
- c_x,c_y
- polygon center coordinatescoordinate units
- R
- circumradiuslength
- x_i,y_i
- Cartesian coordinates of vertex icoordinate units
- K
- shoelace areacoordinate units squared
GRAPH READOUT
Separate object geometry from placement geometry
The plot carries both intrinsic and coordinate-frame information.
Intrinsic edge
-Side length depends only on radius and side count.
Placement envelope
-Bounding width and height respond to rotation.
Coordinate check
-The vertex centroid should return the entered center.
Decision takeaway: Use equal plot scales; otherwise a regular polygon can look stretched while its coordinates remain correct.
EXPORT CHECKLIST
Preserve these conventions with the vertex list
- Angle origin
- Rotation direction
- Coordinate units
- Vertex order
- Center definition
- Closing edge from last to first
Applied decisions
Coordinate tasks served by the graph
Hexagonal bolt pattern
A six-hole pattern is centered away from the global origin and rotated to place one hole on a datum.
What the result clarifies: The vertex table supplies hole centers and the bounding box checks fixture clearance.
Map symbol footprint
A regular marker must rotate without changing its geographic center.
What the result clarifies: The centroid confirms placement while the graph reveals the rotated envelope.
Worked default scenario
Current-input substitution and reconciliation
Key terms
Glossary for interpreting the result
- Circumcenter
- The common center used to generate all polygon vertices.
- Rotation angle
- The angular offset applied to the first vertex before stepping around the polygon.
- Vertex order
- The clockwise or counterclockwise sequence defining the boundary.
- Shoelace formula
- A coordinate cross-sum used to calculate polygon area.
- Bounding box
- The smallest axis-aligned rectangle containing every plotted vertex.
- Closure residual
- The numerical mismatch between the calculated vertex centroid and entered center.
Method references
References for this calculator's specific method
Scope and limitations
Coordinates describe ideal vertices in a Cartesian plane. Screen pixels, map projections, machine compensation, corner radii, and manufacturing tolerances are not included.
Polygon Graphing Calculator | Regular Polygon Coordinates and Rotation FAQ
Where is the first vertex placed?
At the entered rotation angle measured counterclockwise from the positive x-axis.
Why can the bounding box change after rotation?
Different vertices become the extreme x and y points even though side length and area remain fixed.
Is the vertex order clockwise?
The generated order is counterclockwise in a conventional y-up Cartesian plane.
Does moving the center change area?
No. Translation leaves all edge vectors and the area unchanged.