Regular Polygon Calculator
Give this tool a side count and a side length and it returns the full profile of any regular polygon from a triangle up to a 100 sided shape. It reports perimeter, area, the interior and exterior angles, the apothem, and the circumradius together. Every value comes from exact trigonometric formulas rather than lookup tables.
How to find the area of a regular polygon
- Enter the Number of sides, a whole number from 3 to 100.
- Enter the Side length in the second field (any positive number).
- Read the Area and Perimeter cards, highlighted at the top of the grid.
- Check the Interior angle and Exterior angle cards, both shown in degrees.
- Review the Apothem and Circumradius cards, then press Copy result to save all six values.
Examples
Hexagon, 6 sides of length 10
sides = 6, side length = 10
perimeter = 60, area = 259.807621, interior angle = 120, exterior angle = 60, apothem = 8.660254, circumradius = 10
Pentagon, 5 sides of length 6
sides = 5, side length = 6
perimeter = 30, area = 61.937186, interior angle = 108, exterior angle = 72, apothem = 4.129146, circumradius = 5.103905
Octagon, 8 sides of length 4
sides = 8, side length = 4
perimeter = 32, area = 77.254834, interior angle = 135, exterior angle = 45, apothem = 4.828427, circumradius = 5.226252
Frequently asked questions
What is the formula for the area of a regular polygon?
Area equals one half times the perimeter times the apothem. The perimeter is the side count times the side length, and the apothem is the distance from the centre to the middle of a side. For a pentagon of side 6 this gives about 61.937186 square units.
How is the interior angle calculated?
The interior angle equals the side count minus 2, times 180, divided by the side count. Every interior angle of a regular polygon is equal, so a hexagon gives (6 minus 2) times 180 over 6, which is 120 degrees.
What is the exterior angle?
The exterior angle equals 360 divided by the side count, because the exterior angles of any polygon always sum to a full turn. Each interior and exterior angle pair adds to 180 degrees, so an octagon has 45 degree exterior angles.
What is the difference between the apothem and the circumradius?
The apothem runs from the centre to the midpoint of a side, while the circumradius runs from the centre to a vertex. The circumradius is always the longer of the two. The apothem uses the tangent of pi over n and the circumradius uses the sine of pi over n.
Why must the polygon have at least 3 sides?
A closed polygon needs at least three edges to enclose any area. The tool rejects a side count below 3 and also rejects fractional side counts, since a polygon must have a whole number of sides.
Is there an upper limit on the number of sides?
The input allows up to 100 sides. As the count climbs the shape looks more and more like a circle, and the apothem and circumradius converge, but 100 keeps the tool focused on genuine polygons rather than a circle approximation.
What units are the results in?
Lengths, perimeter, apothem and circumradius carry whatever unit you enter for the side. Area is in the matching square unit. Angles are always reported in degrees, not radians.
Does this handle irregular polygons?
No. Every formula assumes all sides and all angles are equal. For an irregular shape you need the coordinates of each vertex, so reach for a polygon area tool that takes a list of points.
Does the calculation happen on a server?
No. The trigonometry runs locally in your browser, so your side count and side length stay on your device and nothing is uploaded or logged.
Learn more
- Regular polygon angles, apothem and area
How side count sets the interior and exterior angles of a regular polygon, and how apothem and circumradius build the area formula.
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