Hi again!!
The shaded sector is a Circular section, and the formulas related to
its perimeter (p) and area (a) are:
p = D + s ,
where 's' is the lengh of the portion of the circunference (arc) that
close the circular section, and relating s to the internal angle in
radians (b) of the circular section:
s = D/2 * b
for the area a, we have:
a = (D/2)^2 * b/2
See the following pages for reference:
"Area And Perimeter Of A Circular Section":
http://www.efunda.com/math/areas/CircleSectionGen.cfm
"Area And Perimeter Of A Circle":
http://www.efunda.com/math/areas/CircleGen.cfm
"Circle" from Wikipedia, the free encyclopedia:
http://en.wikipedia.org/wiki/Circle
Note that if s is the entire circle, then b = 2*PI , then:
s = D/2 * 2*PI = D*PI
and
a = (D/2)^2 * (2*PI)/2 = R^2 * PI
p = D + s = 1 + s
and
p = PI * D = PI
then
PI = 1 + s ==> s = PI - 1 ==>
==> s = D/2 * b = b/2 = PI - 1 ==>
==> b/2 = PI - 1
a = (D/2)^2 * b/2 = (1/2)^2 * (PI - 1) =
= 1/4 * (PI - 1) =
= (PI - 1)/4 =
= 0.5354
Other way is, when we found that b/2 = PI - 1, we continue calculating
the ratio between 2*PI (the angle of the entire circunference) and b
(the angle of the shaded section). This will show which portion (q) of
the circle is shaded:
q = b / 2*PI = b/2 / PI =
= (PI - 1) / PI
The ratio between respective angles are the same that the ratio
between respective areas, in effect:
For circle A = PI * R^2 = PI * D^2/4
For section a = b/2 * R^2
Then
q1 = a/A = (b/2 * R^2) / (PI * R^2) =
= b/2*PI
= q
Then, from a/A = q we have:
a = q * A
= ((PI - 1) / PI) * (PI * D^2/4) =
= (PI - 1) * D^2/4 =
= (PI - 1) * 1/4 =
= 0.5354
I hope this helps you. Please, if you find something unclear or detect
some unintentional mistake, please request for a clarification,i will
gladly respond your requests for further assistance on this topic.
Regards.
livioflores-ga |
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