Full resolution (JPEG) - On this page / på denna sida - Mensuration - To find the area of a segment of a circle - To find the radius corresponding to the arc when the chord and the height of the segment are given
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200 MENSURATION.
Solution
:
Area of segment — A — ^l^L _ 0.433 r2
360
^ = 60X9X9X3.1416 _ 0433X|X9
360
A = 42.4116 — 35.073
A = 7.3386 square inches.
In this example the arc was 60°, consequently the triangle
is equilateral ; therefore its area is found by the formula
0.433 r2
. (See area of equilateral triangles, page 194).
Note.—When the segment is greater than a semicircle,
calculate by preceding rules and formulas the area of the lesser
portion of the circle; subtract it from the area of the whole
circle. The remainder is the area of the segment.
To Find the Radius Corresponding to the Arc, when the
Chord and the Height of the Segment Are Given.
Rule.
Add the square of the height to the square of half the
chord ; divide this sum by twice the height, and the quotient is
the radius. In a formula this may be written
:
r _ n* -f A*
)
v = radius = c b or cf ) ( See Fig. 7).
n = half the chord = db
h = height = df
The above rule and formula may be proved by rules for
right-angled triangles; thus, c b or r equals hypothenuse, and «,
or half the chord, equals perpendicular, and c d}
which is equal
to r — h, is the base. From the rule that the square of the
hypothenuse is equal to the sum of the square of the base and
the square of the perpendicular, we have :
r2 = ;;
2
+ {r — hf
r2 = n2
+ r2 — 2rh + h2
r2
~r2
-\-2rh — n* + h2
2 rh = n2
+ /i
2
Zh
The perpendicular height of the triangle is always equal to
the radius minus the height of the segment. ( See triangle a b c,
and height, df, Fig. 7).
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