How to use the regular polygon calculator
- Enter the number of sides, a whole number from 3 to 1,000.
- Choose which measurement you know: the side length, circumradius, apothem, area or perimeter, and type its value.
- Pick a length unit if you want the answers labeled, or keep plain numbers.
- Read the table of all measurements. Copy it or save it as CSV for your notes or a cutting list.
Formula
side s = 2 R sin(pi / n) = 2 a tan(pi / n)
circumradius R = s / (2 sin(pi / n)) and apothem a = s / (2 tan(pi / n))
perimeter = n x s and area = n x s x a / 2
interior angle = (n - 2) x 180 / n and diagonals = n (n - 3) / 2
The circumradius R is the distance from the center to a corner and the apothem a the distance from the center to the middle of a side. If the area is the known value, the side is found from s = sqrt(4 A tan(pi / n) / n). Angles are in degrees.
A hexagon with 10-unit sides, and others
A regular hexagon with side 10 has circumradius 10, apothem 8.660254, perimeter 60 and area 259.80762. Each interior angle is 120 degrees, there are 9 diagonals, the longest is 20 and the shortest 17.3205.
A pentagon whose apothem is 5 has circumradius 6.18034, side 7.265425 and area 90.8178. An octagon with a perimeter of 40 has sides of 5, a circumradius of 6.532815 and an area of 120.7107. A 12-sided shape with an area of 100 has sides of 2.988585.
Measuring a polygon you will build
Gazebos, planters, tabletops and tiles are often regular polygons. The side length is what you cut, the circumradius is the largest circle the shape fits in, and the apothem is the width across flats divided by two. A hexagon bolt or a hexagonal table is often specified by the distance across flats, which is twice the apothem.
If you know the distance across the corners, divide by two to get the circumradius, and choose that as the known measurement.
Checking the answers
Every polygon is a set of identical triangles around the center, so area equals n times the area of one triangle. The page uses this relation for the area. For an irregular shape with known corner coordinates, use the Polygon Area Calculator, and for the area of simple shapes the Area Calculator.
More layout tools
For layouts and cutting sizes, try the Ellipse String Layout Calculator, Dovetail Layout Calculator and Flying Geese Cutting Calculator.
Assumptions and limits
- The polygon is regular: all sides and all angles are equal.
- The number of sides is a whole number, and the known measurement is greater than zero.
- Results are shown with up to 8 significant digits and are exact for the ideal shape.
- The unit you choose only labels the answers; it does not convert between units.
Frequently asked questions
What is the difference between the apothem and the circumradius?
The circumradius reaches a corner and the apothem reaches the middle of a side. The apothem is always shorter, and for many sides the two are nearly equal.
How many diagonals does a polygon have?
Each corner connects to n - 3 others, and each diagonal is counted from both ends, so there are n (n - 3) / 2. A hexagon has 9 and a square has 2.
Why is a triangle missing the diagonal rows?
A triangle has no diagonals, because every pair of corners is already joined by a side, so the longest and shortest diagonal rows are left out.
Can I get the side from a given area?
Yes. Choose Area as the known measure. The side comes from s = sqrt(4 A tan(pi / n) / n), and the rest follows from the side.