Count all diagonals and find lengths for a chosen vertex step in a regular polygon. The displayed diagonal results includes enough working to inspect signs and scale.
Diagonal counts support network connections and triangulation; diagonal lengths describe regular frames and chord patterns.
An n-gon has n(n−3)/2 diagonals. In a regular polygon, a k-step chord has length s·sin(kπ/n)/sin(π/n).
On this page, the displayed diagonal results follows this definition rather than any similarly named measure.
A regular octagon has 20 diagonals. With side 5, a two-step diagonal is about 9.239.
The step k must run from two through floor(n/2); step one is a side, and larger steps duplicate a shorter route around the polygon. If the Regular Polygon Diagonal assumptions do not fit, consider center geometry.
When regular polygon diagonal looks surprising, restore the sample and vary side length by itself.
Count vertex pairs excluding sides, then use the common circumcircle chord ratio for the selected step.
Write each source value under its matching label before calculating: number of sides, side length and vertex step for length. This preserves the assumptions behind diagonal results and makes a later check possible without reopening the original problem.
Increasing k lengthens the chord until the farthest vertex separation is reached. This relationship remains useful even when the final diagonal results is rounded.
The total count ignores lengths; the step calculation identifies one diagonal family. The two results may share inputs while retaining different meanings.
Before publishing or sharing diagonal results, decide on units, significant digits, and whether an exact form is expected. Those choices belong to interpretation rather than the calculator engine.
When reporting the answer, state the diagonal results first, then its value and unit. Add number of sides, side length and vertex step for length if someone else must verify the work independently.
A chosen diagonal must exceed the side for k above one and cannot exceed twice the circumradius. Symmetric vertex steps describe the same chord family.
For Regular Polygon Diagonal, the best cross-check uses a different identity rather than entering the same formula again. Save enough context to reproduce both routes: field labels, units, angle convention, and unrounded intermediate values. Two independent paths converging on the same quantity make a copied or transposed input easier to detect.
Each diagonal is named from both endpoints.
No, except within the same vertex-step family.
A side, not a diagonal.
Zero.