What makes a polygon regular
A polygon is regular when all of its sides are the same length and all of its angles are equal. An equilateral triangle, a square and a regular hexagon are the familiar cases, but the rule holds for any number of sides. That symmetry is what lets a single measurement unlock everything else: give the calculator the number of sides and one length, and the rest of the shape is fully determined.
The formulas this calculator uses
The angles come straight from the number of sides. Each interior angle is (n minus 2) times 180 divided by n, so a hexagon has 120 degrees and a decagon has 144. Each exterior angle is simply 360 divided by n, and the interior angles always sum to (n minus 2) times 180. For size, the calculator works from the side length s: the perimeter is n times s, the apothem (the distance from the centre to the middle of a side) is s divided by twice the tangent of pi over n, and the area is half the perimeter times the apothem. The circumradius, from the centre out to a corner, is s divided by twice the sine of pi over n.
Side, apothem or circumradius: three ways in
You do not always start from the side. Sometimes you know how far the centre sits from a corner, or from the middle of an edge. The calculator lets you enter any one of the three, converts it back to a side length internally, and then reports every property together. That makes it just as useful for a drafting problem, where you might fit a shape inside a circle of known radius, as for a geometry exercise that hands you the side and asks for the area.
Saturn's hexagon and decagon
Regular polygons are not only a classroom idea. Saturn wears a startlingly clean hexagon of cloud around its north pole, and a ten-sided decagon has been reported over its south. Setting the calculator to six sides and then to ten shows exactly how the interior angle opens from 120 to 144 degrees as corners are added, the same geometric shift that plays out in those planetary jet streams. It is a neat way to connect the tidy formula on the page to a genuinely strange thing happening in the outer solar system.
Frequently asked questions
What is the interior angle of a regular polygon?
It is (n minus 2) times 180, divided by n, where n is the number of sides. A hexagon gives 120 degrees and a decagon gives 144 degrees.
How do you find the area of a regular polygon?
Multiply half the perimeter by the apothem, the distance from the centre to the middle of a side. Equivalently, area equals one quarter of n times the side squared times the cotangent of pi over n.
What is the difference between the apothem and the circumradius?
The apothem runs from the centre to the middle of a side, and the circumradius runs from the centre to a corner. The circumradius is always the larger of the two.
What is the interior angle of a hexagon and a decagon?
A regular hexagon has interior angles of 120 degrees; a regular decagon has 144 degrees. Both follow the (n minus 2) times 180 over n rule.
Do the angles of a polygon add up to 360 degrees?
The exterior angles always add up to 360 degrees, for any number of sides. The interior angles add up to (n minus 2) times 180, which grows as sides are added.
Is a circle a polygon with infinitely many sides?
Not formally, but it is a useful intuition. As you add sides to a regular polygon, it hugs its circumscribed circle more and more closely, and its area approaches that of the circle.