Machinery's Handbook, 31st Edition
CIRCLE 49 Circle.— A circle is the set of points equidistant from a given point in the plane. Another name for this set of points is locus , which is a curve formed by all the points satisfying a particular equation. The general form for the equation of a circle is x 2 + y 2 + 2 gx + 2 fy + c = 0, where - g and - f are the coordinates of the center and the radius is r g 2 f 2 c = + – . The standard form of a circle (center-radius form) is x h – ( ) 2 y k – ( ) 2 + r 2 = Y
where r = length of the radius and point ( h , k ) is the center. When the center of circle is at point (0, 0), the equation reduces to x 2 y 2 + r 2 = or r x 2 y 2 = + Example 1: Point (4, 6) lies on a circle whose center ( h, k ) is at point ( - 2, 3). Find the circle’s equation. Solution: The radius is the distance from the center point ( - 2, 3) to point (4, 6), found using the method of Example 1 on page 43. r 4 –2 ( ) – [ ] 2 6– 3 ( ) 2 + 6 2 3 2 + = =
Center ( h, k )
r
X
=
45
Using the form x h – ( ) 2 y k – ( ) 2 + r 2 = x +2 ( ) 2 y – 3 ( ) 2 + x 2 4 x 4 y 2 6 y – 9 + + + + 45 = = x 2 y 2 4 x 6 y – – 32 + + = 0 and substituting h = –2, k = 3, and r 2 = 45:
x h – ( ) 2 y k – ( ) 2 + r 2 = x +2 ( ) 2 y – 3 ( ) 2 + x 2 4 x 4 y 2 6 y – 9 + + + + 45 = = x 2 y 2 4 x 6 y – – 32 + + = 0
Additional Formulas: Listed below are additional formulas for determining the geome try of plane circles and arcs. Although trigonometry and circular measure are related, they deal with angles in entirely different ways. In each of these formulas, the entered measure of the angles are in degrees, and the formulas convert them to radian measure (see page 94, in TRIGONOMETRY ). C = π D 2 π R = Radius R = N π -- X 2 Y 2 = + Diameter D = diameter of circle = 2 R C π = -- Area A = π R 2 X = R 2 Y 2 – Y = R 2 X 2 – Area of complement sector M R 2 π R 2 4 – ----- 0.2146 R 2 = =
I = distance to section T H = height of section T Q = chord length for segment S P = chord length for segment section T T + S = area of segment = R 2 sin –1 P
Fig. 6a.
2 R ---- IP 2 – ---
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