Free · Regular star geometry

Regular Star Boundary Lengths and Filled Area

Calculate linked dimensions for regular five-, six-, seven- and eight-point stars with their distinct perimeter and area boundaries explained.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

For an octagram with first-boundary edge b=1, P=16 and outermost filled area≈32.9705627484771.

First boundary P=2nb. For seven/eight points, outer area=n a²cot(π/n)/4+n a√(d²−a²/4)/2.

Worked example

Enter these known values and leave the other values blank.

Shape
Octagram
b
1
a
1.8478
c
1.4142
l
3.414
d
2.414
Perimeter
16
Area
32.97

Assumptions and limitations

  • Each option is a particular regular construction with one positive scale. Linked lengths are not arbitrary independent rays; clear an unknown before solving backwards.
  • There are no unit selectors in the captured model. Use the same length unit throughout. Perimeter has that length unit, and area has its square.
  • P always measures the blue first star boundary. Five/six-point A fills that boundary. Seven/eight-point A fills the amber outermost boundary, whose edge is d; its perimeter would be2nd, which is not the displayed P.
  • The original relationships and diagram labels are retained: a joins adjacent first-boundary tips, b is a first-boundary edge, c is a central polygon side, l=2b+c, and d is a seven/eight-point outer edge. The diagram is a fixed schematic, not a plot of entered values.
  • Five/six-point area inverses choose the positive square root. Seven/eight-point areas remain output-only as captured; no new area-entry inverse is added. Inconsistent independently entered linked dimensions are not a different regular star.

Common questions

Why do perimeter and area use different outlines for seven/eight points?

The captured construction includes two boundaries. P follows the first outline through edge b; A covers the outermost outline through edge d. The diagram and labels distinguish them.

Can I choose a different ray length independently?

No. Each regular option has fixed length ratios and one positive scale. Enter one known dimension or a supported area inverse.

References

Correct the adjacent first-tip chord of the octagram to a=2b cos(pi/8), in both directions. Count each of the seven/eight outer triangles as base×height/2 instead of the captured half contribution. Preserve the first-boundary perimeter and outermost-boundary area distinction, all other equations/inverses and output-only seven/eight-point area controls. Retain every captured strictly positive length, perimeter and area domain, testing the sign directly to avoid treating tiny positive values as zero. All four geometrically similar constructions have one positive scale; zero/negative scale is not a regular star.

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