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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
-A----
--A---
-A--- R--- R---
-S-- S--
-r--
-r-
Sides S2
R2 r2 S
r R r R S
3 0.4330 1.2990 5.1962 0.5774 2.0000 1.7321 3.4641 0.5000 0.2887 4 1.0000 2.0000 4.0000 0.7071 1.4142 1.4142 2.0000 0.7071 0.5000 5 1.7205 2.3776 3.6327 0.8507 1.2361 1.1756 1.4531 0.8090 0.6882 6 2.5981 2.5981 3.4641 1.0000 1.1547 1.0000 1.1547 0.8660 0.8660 7 3.6339 2.7364 3.3710 1.1524 1.1099 0.8678 0.9631 0.9010 1.0383 8 4.8284 2.8284 3.3137 1.3066 1.0824 0.7654 0.8284 0.9239 1.2071 9 6.1818 2.8925 3.2757 1.4619 1.0642 0.6840 0.7279 0.9397 1.3737 10 7.6942 2.9389 3.2492 1.6180 1.0515 0.6180 0.6498 0.9511 1.5388 12 11.196 3.0000 3.2154 1.9319 1.0353 0.5176 0.5359 0.9659 1.8660 16 20.109 3.0615 3.1826 2.5629 1.0196 0.3902 0.3978 0.9808 2.5137 20 31.569 3.0902 3.1677 3.1962 1.0125 0.3129 0.3168 0.9877 3.1569 24 45.575 3.1058 3.1597 3.8306 1.0086 0.2611 0.2633 0.9914 3.7979 32 81.225 3.1214 3.1517 5.1011 1.0048 0.1960 0.1970 0.9952 5.0766 48 183.08 3.1326 3.1461 7.6449 1.0021 0.1308 0.1311 0.9979 7.6285 64 325.69 3.1365 3.1441 10.190 1.0012 0.0981 0.0983 0.9988 10.178
A = NS2 cot α ÷ 4
= NR2 sin α cos α
= Nr2 tan α
r = R cos α
= (S cot α) ÷ 2 =
(A cot α) ? N
R = S ÷ (2 sin α)
= r ÷ cos α =
A ? (N sin α cos α)
S = 2R sin α
= 2r tan α = 2
(A tan α) ? N
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