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MENSURATION. 1 99
To Find the Length of Arc of a Segment of a Circle.
The length of the arc may be calculated by the formula,*
/__ 8 c — C
3
/= Length of arc, a fb \
c == Length of chord from a tof v ( See Fig. 7).
C = Length of chord from a to b )
Rule.
Multiply the length of the chord of half the arc by
8 ; from the product subtract the length of the chord of the arc ;
divide the remainder by 3, and the quotient is the length of
the arc.
When chord and height of segment are known, the chord of
half the arc is calculated thus
:
Chord of half the arc = ,y/ wa _j_ ^2
h = Height of segment (see df, Fig. 7).
;/ = Half the length of chord ( see a d or b d, Fig. 7).
When only the radius and the height of the segment are
known, the length of the chord of the whole arc expressed in
these terms will be : 2 X \/ 2 r h /i’
2
The chord of half the arc will be :
y/2 rh
Therefore the length of the arc will be
:
_ 8 X s/lrh— 2 X V2 r h — h2
1 — 3
/= length of arc (afb, Fig. 7).
h = height of segment ( df, Fig. 7 ).
r = radius of circle {cf, Fig. 7).
To Find the Area of a Segment of a Circle.
(See Fig. 7).
Ascertain the area of the whole
sector and from this area subtract the
area of the triangle, and the rest is the
area of the segment.
Example.
Find the the area of the segment
when the radius is 9 inches and the
arc 60°.
* This formula is called " Huyghens’s approximate formula for circular arcs,"
but it is so close that it may for any practical purpose be considered absolutely cor-
rect for arcs having small center angles; for center angles as large as 120 °
, the
result is only one quarter of one per cent, too small, and even for half a circle the
result is scarcely more than one per cent, small as compared to results calculated by
taking ~ as 3.1416.
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