Machinery's Handbook, 31st Edition
82
POLYGONS
Fig. 5. Hexagons: (a) Inscribed (Drawn Inside) a Circle; (b) Circumscribed (Drawn Around) a Circle. Formulas and Table for Regular Polygons: The following formulas and table can be used to calculate area, side length, and radii of inscribed and circumscribed polygons on a circle. A nS nR nR r R S A n = = = = = = 2 2 2 4 2 cot sin cos tan cos cot α α α α α α , nR nR r R S A = = = = = 2 2 sin cos tan cos cot cot α α α α α α α ,
2
α
cot
nS
cot
A
=
n
4
2
r
A
S =
A
α
tan
S
= r
A
r A
α
tan tan
=
,
R
= S R 2 tan = r 2
=
α
=
α
=
sin α
2
2
,
R
= S R 2
=
=
α
=
sin
2
N
N
α α cos N
α α
2 α α cos
sin
sin cos
N
α α
2
sin
sin cos
Area, Length of Side, and Inscribed and Circumscribed Radii of Regular Polygons No. of Sides, n A S 2 --- A R 2 --- A r 2 -- R S -- R r -- S R -- S r -- r R -- r S --
3 0.4330 1.2990 5.1962 4 1.0000 2.0000 4.0000 5 1.7205 2.3776 3.6327 6 2.5981 2.5981 3.4641 7 3.6339 2.7364 3.3710 8 4.8284 2.8284 3.3137 9 6.1818 2.8925 3.2757 7.6942 2.9389 3.2492
0.5774 2.0000 1.7321 3.4641 0.5000 0.2887 0.7071 1.4142 1.4142 2.0000 0.7071 0.5000 0.8507 1.2361 1.1756 1.4531 0.8090 0.6882
1.0000 1.1547 1.0000 1.1547 0.8660 1.1524 1.1099 0.8678 0.9631 0.9010 1.3066 1.0824 0.7654 0.8284 0.9239 1.4619 1.0642 0.6840 0.7279 0.9397 1.6180 1.0515 0.6180 0.6498 0.9511 1.9319 1.0353 0.5176 0.5359 0.9659 2.5629 1.0196 0.3902 0.3978 0.9808 3.1962 1.0125 0.3129 0.3168 0.9877 3.8306 1.0086 0.2611 0.2633 0.9914 5.1011 1.0048 0.1960 0.1970 0.9952 7.6449 1.0021 0.1308 0.1311 0.9979
0.8660 1.0383 1.2071 1.3737 1.5388 1.8660 2.5137 3.1569 3.7979 5.0766 7.6285
10
12 11.196 16 20.109 20 31.569 24 45.575 32 81.225 48 183.08 64 325.69
3.0000 3.2154 3.0615 3.1826 3.0902 3.1677 3.1058 3.1597 3.1214 3.1517 3.1326 3.1461
3.1365 3.1441 10.190 1.0012 0.0981 0.0983 0.9988 10.178 Example 1: A regular hexagon is inscribed in a circle of 6 in. diameter. Find the area and the radius of an inscribed circle. Here, R = 3 in. and n = 6. From the table, area A = 2.5981 R 2 = 2.5981 × 9 = 23.3829 in 2 . Radius of inscribed circle, r = 0.866 R = 0.866 × 3 = 2.598 in. Example 2: An octagon is inscribed in a circle of 100 mm diameter. Thus, R = 50 mm and n = 8. Find the area and radius of an inscribed circle. A = 2.8284 R 2 = 2.8284 × 2500 = 7071 mm 2 = 70.7 cm 2 . Radius of inscribed circle, r = 0.9239 R = 09239 × 50 = 46.195 mm. Example 3: Thirty-two bolts are to be equally spaced on the periphery of a 16-in. diameter bolt-circle. Find the chordal distance between the bolts. Chordal distance equals the side length S of a polygon with n = 32 sides and R = 8. Hence, S = 0.196 R = 0.196 × 8 = 1.568 in. Example 4: Sixteen bolts are to be equally spaced on the periphery of a bolt-circle, 250 mm diameter. Find the chordal distance between the bolts. Chordal distance equals the side length S of a polygon with 16 sides. R = 125. Thus, S = 0.3902 R = 0.3902 × 125 = 48.775 mm.
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