An optional helper. The facts here are standard geometry, worked out from each polygon itself; nothing on this page knows about any puzzle. It runs entirely in your browser, and your notes stay on this device.
Pick a polygon
Step 1 joins each corner to the next, making an ordinary polygon. A larger step joins every second, third… corner, making a star: 5 sides with step 2 is the pentagram {5/2}. When sides and step share a factor, the result is several polygons laid over each other: {6/2} is two triangles.
Pentagon
“Diagonals” draws every line between two corners that aren't neighbours, coloured by how many steps apart the corners are. “Crossing points” marks where diagonals cross inside a regular polygon.
Facts
Every length and area below is for this size. Tap + to copy a fact to your notepad.
Which polygon?
Know one fact and want the polygon? Pick the fact, type its value, and every regular polygon or star that fits is listed. Angles are matched to within 0.01°.
Notepad
Saved in this browser only. Tap a note's value to put it in the last box you used below.
Nothing noted yet.
Quick coordinate helpers
Digits go in order: degrees, then minutes with three decimals. 14 or 15 digits make a full coordinate; 7 or 8 make one half.
Takes decimal degrees, or degrees minutes seconds (N 45 12 20.7 W 93 54 19.3), and writes them the geocaching way.
Where the facts come from
Angles, radii, perimeters and areas use the standard formulas for regular polygons and star polygons {n/d}. A star's “area inside the outline” is the area of the region it encloses. Its “area counted by winding” counts the middle again for each extra time the outline goes round, which is what the shoelace formula gives when you trace the star.
The number of points where the diagonals of a regular polygon cross uses Bjorn Poonen and Michael Rubinstein's 1998 formula. It was checked here against a direct count for every polygon up to 120 sides, and it counts a point where three or more diagonals meet only once. The number of regions the diagonals make was counted directly, so it is given up to 120 sides.
A regular polygon can be drawn with compass and straightedge exactly when its number of sides is a power of 2 times distinct Fermat primes (3, 5, 17, 257, 65537), the Gauss–Wantzel theorem. The number of ways to cut a polygon into triangles with diagonals that don't cross is a Catalan number.