A regular polygon is fixed by its number of sides and any one measurement: the side length, the apothem or the circumradius. Give this calculator those two facts and it returns everything else: area, perimeter, all three angles, both radii, both widths and a drawing of the shape. If your polygon is irregular, switch to coordinate mode and type the corner points in order; the calculator finds the area and perimeter from the coordinates themselves.
How to use this polygon calculator
- Regular polygon: leave the type on “Regular polygon”, or tap a quick shape (hexagon and octagon are the two most searched for). Type the number of sides if your shape is not listed.
- Choose the measurement you know: a side length, the apothem (the distance from the centre straight out to the middle of a side), or the circumradius (the distance from the centre to a corner). Enter its value, pick the unit, and press Calculate.
- Irregular polygon: switch the type to “Irregular polygon (corner coordinates)”. Type each corner as x, y, one per line, walking around the shape in order, clockwise or anticlockwise, without jumping across it. The calculator works out the area and perimeter and checks that no edges cross.
The formulas
Everything about a regular polygon comes from one angle: the half central angle, π/n. From the side length s, the two radii are:
The area can be written three equivalent ways; the calculator uses the first and the others are useful checks:
Two widths matter in workshops. The width across flats (flat side to flat side) is 2a, twice the apothem; the width across corners is 2R, twice the circumradius. A hex bolt head is measured across flats, because that is the spanner size, while a hexagonal tile pattern is often laid out across corners. The results table prints both so the two are never confused.
Irregular polygons: the shoelace formula
For a shape given by corner coordinates (x₁, y₁), (x₂, y₂), … listed in order around the shape, the area is half the absolute value of the sum of cross products of consecutive corners:
The name comes from the criss-cross pattern the multiplication makes when written out, like lacing a shoe. It works for any simple polygon, convex or with dents, as long as the corners are listed in order around the boundary. The perimeter is simply the sum of the edge lengths, including the closing edge from the last corner back to the first.
Worked examples
Example 1: a hexagon with side 6 in
Regular hexagon, n = 6, side 6 in. Circumradius = 6 / (2 sin 30°) = 6.00 in (on a hexagon the radius equals the side, which is why hexagons tile so neatly). Apothem = 6 / (2 tan 30°) = 5.20 in, so the width across flats is 10.39 in and across corners 12.00 in. Area = 6 × 36 / (4 tan 30°) = 93.53 sq in. Perimeter = 36.00 in, interior angle 120.00°, diagonals 9.
Example 2: an octagon with side 4 ft
Regular octagon, n = 8, side 4 ft, the shape of a gazebo floor. Area = 8 × 16 / (4 tan 22.5°) = 77.25 sq ft. Apothem = 4.83 ft (across flats 9.66 ft), circumradius = 5.23 ft (across corners 10.45 ft), perimeter = 32.00 ft, interior angle 135.00°, diagonals 20. If you only knew the width across flats of an existing octagon, halve it to get the apothem, 4.83 ft, and enter that instead; you will land on the same side length and area.
Example 3: an L-shaped plot from coordinates
An L-shaped garden bed has corners, in metres and in order: (0, 0), (6, 0), (6, 2), (2, 2), (2, 4), (0, 4). The shoelace sum gives an area of 16.00 sq m, which matches splitting the L into a 6 by 2 rectangle (12) plus a 2 by 2 square (4). The perimeter, including the closing edge, is 20.00 m. This is the case preloaded in coordinate mode, so you can press Calculate there and see the same numbers.
Reading your results
Area and perimeter are the headline figures: material to cover the shape, and edging or trim to go around it.
Apothem and circumradius answer two different questions. The apothem is the radius of the largest circle that fits inside the polygon (the incircle); the circumradius is the radius of the circle that passes through every corner (the circumcircle). If you are cutting a polygon from a round blank, the blank must be at least the across-corners width, 2R. If you are fitting a polygon inside a round hole, the hole must clear the same 2R.
Interior angle is the angle inside each corner, the number you set on a mitre saw (halved) when cutting frame or trim pieces that meet at the corners. Exterior angle is the turn you make walking around the shape, and the central angle is the slice angle at the centre; for regular polygons the exterior and central angles are equal.
Diagonals counts the straight lines joining non-adjacent corners. It matters for bracing and for star patterns drawn inside the shape, and it grows fast: a decagon already has 35.
Common mistakes
- Confusing the apothem with the circumradius. They differ by a factor of cos(π/n). On an octagon that is about 8 percent, enough to spoil a cut. The apothem goes to the middle of a side; the circumradius goes to a corner.
- Typing the across-flats width as the side. A hex head marked 10 mm is 10 mm across flats, so its apothem is 5 mm and its side is 5.77 mm. Enter the apothem, not the width, into the side box.
- Listing coordinate points out of order. The shoelace formula needs the corners in boundary order. Points typed in a zigzag make edges cross, and the calculator flags it instead of printing a wrong area.
- Using regular formulas on an irregular shape. If the sides are not all equal, the regular mode does not apply; measure the corners and use coordinate mode.
- Expecting a polygon area from a diameter alone. A “diameter” for a polygon is ambiguous: across flats (2a) or across corners (2R). Halve whichever you have and enter it as apothem or circumradius.
Limitations
- Regular mode assumes every side and every angle is equal. Real shapes that are nearly regular but not exact belong in coordinate mode.
- Coordinate mode assumes a simple polygon: edges meet only at shared corners. If the edges cross, the calculator reports the crossing instead of an area, because a crossed (bowtie) outline does not enclose one region.
- Coordinate mode takes up to 50 corners, in one unit, on a flat grid. It does not handle curved boundaries, holes inside the shape, or latitude and longitude coordinates, where the grid itself is distorted.
- All results are shown rounded to two decimal places; the underlying calculation keeps full precision.
Frequently asked questions
What is the apothem of a polygon?
The apothem is the distance from the centre of a regular polygon to the middle of one side, measured at a right angle to that side. It is also the radius of the polygon’s incircle. Every regular polygon has one; irregular polygons do not have a single apothem.
How do I find the area of a hexagon?
For a regular hexagon with side s, the area is (3√3 / 2) s², about 2.598 s². With side 6 that is 93.53 square units, exactly what Example 1 works through. In the calculator, tap Hexagon, enter the side, and read the area.
How do I find the area of an octagon?
For a regular octagon with side s, the area is 2(1 + √2) s², about 4.828 s². If you know the width across flats instead, enter half of it as the apothem; the calculator recovers the side and the area from there.
Can this calculator do irregular polygons?
Yes, in coordinate mode. Type the corners in order around the shape and it applies the shoelace formula, adds up the perimeter, plots the outline and checks that no edges cross. It handles shapes with dents (concave corners) as well as convex ones.
What is the difference between inradius and circumradius?
Two names for the two radii in the results: the inradius is the apothem (centre to a side), and the circumradius is centre to a corner. The circumradius is always the larger of the two; they are equal only in the limit of a circle.
How many sides can I enter?
From 3 (a triangle) up to 100 in regular mode. As the number of sides grows, the polygon approaches a circle and its area approaches πR², which is a useful sanity check on very large inputs.
Related calculators
- Math calculators
- Triangle Calculator: the three-sided case with its own angle and side rules.
- Circle Calculator: the shape a many-sided polygon approaches.
- Rectangle and Square Calculator: four-sided shapes where the sides need not all be equal.
- Area and Volume Calculator: areas of plots and spaces in everyday units.
Method: regular polygon geometry, A = n s² / (4 tan(π/n)), radii from the half central angle π/n; irregular area by the shoelace formula over ordered vertices, with a simple-polygon (non-crossing) check. Figures checked against independently calculated fixtures. Last reviewed: 20 October 2026.
Disclaimer: This calculator gives estimates for general information only. It is not financial, medical or other professional advice.