Formulas and Table for Regular Polygons.—A
regular polygon is a many-sided, two-
dimensional figure in which the lengths of the sides are equal. Thus, the angle measures are
also equal. An equilateral (equiangular) triangle is the polygon with the least number of
sides. The following formulas and table can be used to calculate the area, length of side, and
radii of the inscribed and circumscribed circles of regular polygons.
where N= number of sides; S= length of side; R = radius of circumscribed circle; r =
radius of inscribed circle; A = area of polygon; and, a= 180° ÷ N = one-half center angle of one
side. See also Regular Polygon on page 74.
Area, Length of Side, and Inscribed and Circumscribed Radii of Regular Polygons
Example 1: A regular hexagon is inscribed in a circle of 6 inches diameter. Find the area and the
radius of an inscribed circle. Here R = 3. From the table, area A = 2.5981R2 = 2.5981
× 9 = 23.3829 square inches. Radius of inscribed circle, r = 0.866R = 0.866 × 3 = 2.598
inches.
Example 2: An octagon is inscribed in a circle of 100 mm diameter. Thus R = 50. Find the area and
radius of an inscribed circle. A = 2.8284R2 = 2.8284 × 2500 = 7071 mm2 = 70.7 cm2. Radius of
inscribed circle, r = 0.9239R = 09239 × 50 = 46.195 mm.
Example 3: Thirty-two bolts are to be equally spaced on the periphery of a bolt-circle, 16 inches
in diameter. Find the chordal distance between the bolts. Chordal distance equals the side S of a
polygon with 32 sides. R = 8. Hence, S = 0.196R = 0.196 × 8 = 1.568 inch.
Example 4: Sixteen bolts are to be equally spaced on the periphery of a bolt-circle, 250
millimeters diameter. Find the chordal distance between the bolts. Chordal distance equals the side
S of a polygon with 16 sides. R = 125. Thus, S = 0.3902R = 0.3902 × 125 = 48.775 millimeters.
No. of