Regular Polygon Calculator
Calculate every measurement of a regular polygon from any one of them: side length, area, perimeter, apothem, circumradius, interior and exterior angles, and diagonal count.
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One measurement determines all the others
A regular polygon with n sides has a single degree of freedom in its size, so knowing the side length, the area, the apothem or the circumradius fixes everything else. This is what makes the calculation useful in practice: a fabrication drawing might specify the distance across flats, a bolt catalogue specifies the distance across corners, and a material estimate needs the area, and converting between them by hand needs a different formula each time.
Across flats and across corners are different measurements
For a hexagonal nut, the distance across flats is twice the apothem and the distance across corners is twice the circumradius, and for a hexagon they differ by a factor of about 1.155. Confusing them is a common and expensive mistake in mechanical work, because a spanner sized by one measurement will not fit a nut specified by the other. Both are given here with explicit names rather than a single ambiguous diameter.
Approaching a circle
As the number of sides grows the polygon converges on its circumscribed circle, and the area approaches pi times the radius squared. Archimedes used exactly this to bound pi, working up to a 96 sided polygon inscribed and circumscribed to get pi between 223/71 and 22/7. The ratio of the polygon area to the circle area is shown below for the same reason: it says how good an approximation the polygon is, which matters when a curve is being approximated by straight segments in manufacturing or graphics.
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Frequently Asked Questions
What is the apothem?
The perpendicular distance from the centre to the middle of a side, which is the radius of the inscribed circle. Twice the apothem is the distance across flats, which is how hexagonal nuts and spanners are usually sized.
What is the difference between across flats and across corners?
Across flats is twice the apothem and across corners is twice the circumradius. For a hexagon they differ by about fifteen percent, which is why a spanner specified one way will not fit a nut specified the other.
Do the interior angles depend on the size?
No. They depend only on the number of sides: each interior angle is 180 times (n−2) divided by n. The exterior angles always sum to 360 degrees regardless of how many sides there are, which is the fact that makes turtle graphics work.
How many diagonals does a polygon have?
n times (n−3) divided by 2. Each vertex connects to all but itself and its two neighbours, and dividing by two avoids counting each diagonal from both ends.
Why does a polygon approach a circle?
Because each side subtends a smaller angle as the count grows, so the boundary deviates less from the circumscribed circle. Archimedes used a 96 sided polygon to bound pi between 223/71 and 22/7 using exactly this convergence.
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How to Use
Choose the number of sides and enter any one measurement to get all the others.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
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