SOLUTION: how do you find the perimeter of a shaded region of a circle the shaded region is 140 degrees the area of the shaded region is 126π

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Question 979530: how do you find the perimeter of a shaded region of a circle
the shaded region is 140 degrees
the area of the shaded region is 126π
Found 4 solutions by josgarithmetic, Edwin McCravy, josmiceli, AnlytcPhil:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Two radii and the length of the arc with this radius.

%28140%2F360%29pi%2Ar%5E2=126pi
The fraction of the full-circle's area given to be equal to 126pi.

%287%2F18%29pi%2Ar%5E2=126pi
r%5E2=126pi%2818%2F7%29%2Fpi
r%5E2=%2818%5E2%29
highlight%28r=18%29

The length of the arc, 2pi%2Ar%28140%2F360%29=%282%2A7%2F18%29pi%2Ar=%282%2A7%2F18%29%2A18%2Api=highlight%2814pi%29.

The perimeter of this sector is r%2Br%2B14pi=18%2A2%2B14pi=highlight%28highlight%2836%2B14pi%29%29.

Answer by Edwin McCravy(20060) About Me  (Show Source):
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
If you can find the radius, then
the problem is solved, since
Circumference = +2%2Api%2Ar+
--------------------------
The whole circle is +360+ degrees of arc
+140+%2F+360+=+7%2F18+
Let +A+ = the area of the whole circle
I can say:
+%28+7%2F18+%29%2AA+=+126%2Api+
+A+=+%28+18%2F7+%29%2A126%2Api+
+A+=+324%2Api+
I know that:
+A+=+pi%2Ar%5E2+ ,so
+pi%2Ar%5E2+=+324%2Api+
+r%5E2+=+324+
+r+=+18+
and
+C+=+2%2Api%2Ar+
+C+=+2%2Api%2A18+
+C+=+36%2Api+
-----------------
The perimeter of the shaded rgion is:
+%28+7%2F18+%29%2A36%2Api+
+14%2Api+ answer

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!

The whole figure is a sector of a circle. The area of a sector of a circle
is given by the formula:

A%22%22=%22%22expr%281%2F2%29theta%2Ar%5E2 where q is in radians.

We convert q = 140° to radians:

%22140%B0%22%2Api%2F%22180%B0%22%22%22=%22%227pi%2F9

126pi%22%22=%22%22%28expr%281%2F2%29%2A7pi%2F9%29%2Ar%5E2

126pi%22%22=%22%22%287pi%2F18%29%2Ar%5E2

Multiply both sides by 18

2268pi%22%22=%22%227pi%2Ar%5E2

Divide both sides by 7p

324%22%22=%22%22r%5E2

18%22%22=%22%22r

18%22%22=%22%22r

The length of the arc is found by the formula

s%22%22=%22%22r%2Atheta where q is in radians.

s%22%22=%22%2218%2A%287pi%2F9%29

s%22%22=%22%2214pi

The perimeter is made up of two radii and the arc at the top:

Perimeter = 36%2B14pi

Edwin