DIMENSIONS OF PLANE FIGURES Machinery's Handbook, 31st Edition
80
Spandrel or Fillet:
The shaded region is the spandrel (fillet).
A r 2 π
r 2
r
c
4 – ---- 0.215 r 2 =
0.1075 c 2
Area
= =
=
Example: Find the area of a spandrel, the radius of which is 0.7 inch. A 0.215 r 2 0.215 0.7 2 × 0.105in 2 = = = Example: If chord c were given as 2.2 inches, what would be the area? A 0.1075 c 2 0.1075 2.2 2 × 0.520 in 2 = = = Parabola: Area A 2 ⁄ 3 xy = =
The area of the shaded portion is equal to two-thirds of a rectangle which has x for its base and y for its height. Example: Let x in the illustration be 15 centimeters, and y be 9 centimeters. Find the area of the shaded portion of the parabola. A 2 ⁄ 3 xy 2 ⁄ 3 15 × 9 × 10 9 × 90cm 2 = = = =
Parabola:
2 x p --- 1
p 2 --
2 x p + ---
2 x p ---
1 2 x p + + ---
l = When x is small in proportion to y , the following is a close approximation: length of arc + ln =
l
y
p x 2
l 4 3 -- x 2 = + Example: If x = 2 feet and y = 24 feet, what is the approximate length l of the parabolic curve? l y 1 2 3 -- x y y 1 = + 2 3 -- x y - 2 2 5 -- – x y - 4 or l y 2 - 2 2 5 – -- x y - 4 + -- 2 24 --- 2 2 5 – -- 2 24 --- 4 =
24 1 2 3 + = 24 1.0046 × 24.04 ft = =
Segment of Parabola: D F E C G B
Area BFC A = = × If FG is the height of the segment, measured at right angles to BC , then: Area of segment BFC 2 ⁄ 3 ( BC )( FG ) = 2 ⁄ 3 Area of parallelogram BCDE
Example: Suppose the length of the chord BC = 19.5 inches, and the distance between lines BC and DE , measured at right angles to BC , is 2.25 inches. Find the area of the segment. Area A 2 ⁄ 3 19.5 × 2.25 × 29.25 in 2 = = = = 2 ⁄ 3 ( BC )( FG )
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