Regular Polygon Incircle (Apothem) Calculator

Find the inscribed circle radius (apothem) of a regular polygon from its number of sides and side length. Also shows the incircle area, polygon area, and their ratio with a diagram.

Input

Enter the number of sides and the side length to find the inscribed circle radius (apothem) of a regular polygon.

Number of sides n

Side length a

Result

Inradius (apothem)

8.660254

Incircle area

235.619449

Polygon area

259.807621

Area ratio (incircle to polygon)

90.689968 percent

Circumradius

Lengths share the input unit and areas use squared units.

How it works

The inscribed circle radius (apothem) is the distance from the center to the midpoint of each side. For side length a and number of sides n, it equals a divided by 2 tan of pi over n.
The incircle area is pi times r squared, and the regular polygon area is n times a squared divided by 4 tan of pi over n. The area ratio is the incircle area divided by the polygon area, approaching 1 (100 percent) as the number of sides increases.
The circumscribed circle radius equals a divided by 2 sin of pi over n, the radius of the circle through every vertex. The inradius (apothem) is always smaller than this.
Enter an integer of 3 or more for the number of sides and a positive value for the side length. Lengths share the input unit and areas use squared units.

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