Machinery's Handbook, 31st Edition
POLYGONS
81
Hyperbola:
ab 2 ---
xy 2 ---
x a -- y b + --
Area BCD A
= =
– ln
Example: The half-axes a and b are 3 and 2 inches, respectively. Find the area shown shaded in the illustration for x = 8 inches and y = 5 inches. Inserting the known values in the formula: Area A 8 5 × 2 ------ 3 2 × 2 ------ 8 3 -- 5 2 + -- ln × – 20 3 (5.167) ln × – = = = 20 3 1.6423 × – 20 – 4.927 15.073 in 2 = = =
b
D
y
B
C
a
x
Ellipse:
Area π ab 3.1416 ab = = = An approximate formula for the perimeter is Perimeter P 3.1416 2 a 2 b 2 + ( ) = = A
a
b
a b – ( ) 2 2.2 – ----------
A closer approximation is P
3.1416 2 a 2 b 2 + ( )
=
Example: The larger, or major, axis is 200 millimeters. The smaller, or minor, axis is 150 millimeters. Find the area and the approximate circumference. Here, then, a = 100, and b = 75. A 3.1416 ab 3.1416 100 × 75 × 23,562 mm 2 235.62 cm 2 = = = = P 3.1416 2 a 2 b 2 + ( ) 3.1416 2 100 2 75 2 + ( ) 3.1416 2 15,625 × = = = 3.1416 31,250 3.1416 176.78 × 555.37 mm = = = = 55.537 cm Polygons.— A polygon is a many-sided figure in a two-dimensional plane. A polygon is sometimes referred to as an n -gon, where n is the number of sides. Triangles are polygons with the least number of sides ( n = 3), followed by quadrilaterals ( n = 4), pentagons ( n = 5), hexagons ( n = 6), heptagons ( n = 7), octagons ( n = 8), and so on. A regular polygon has congruent sides (all sides are of equal measure) and, hence, its interior angles are congruent. In Fig. 4a, β is the measure of each interior angle, 180 – β is the exterior angle measure at each vertex. α is a measure of the central angle. Irregular polygons (Fig. 4b) are polygons whose sides are not all congruent. Both Fig. 4a and Fig. 4b show convex polygons, in which all exterior angles measure less than 180 degrees. A concave polygon has some interior angles that measure greater than 180 degrees (see Fig. 4c). All the formulas in this section concern convex regular polygons.
Fig. 4. Polygons: (a) Convex Regular; (b) Convex Irregular; (c) Concave Irregular. Fig. 5 shows how a regular polygon is either inscribed (drawn inside) or circumscribed (drawn around) a circle. A polygon inscribed within a circle, as shown in Fig. 5a, is drawn so that all its vertices touch the circle. Its radius is marked r . A polygon that circumscribes a circle, as shown in Fig. 5b, is drawn so that the circle touches each of the sides of the polygon. Its radius is marked R .
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