SOLUTION: Three circles of radii 8.5, 11.9, and 15.7 cm are mutually tangent. Find the area bounded by the three circles. area =

Question 1043961: Three circles of radii 8.5, 11.9, and 15.7 cm are mutually tangent.
Find the area bounded by the three circles.
area =
Found 2 solutions by Alan3354, advanced_Learner:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
Three circles of radii 8.5, 11.9, and 15.7 cm are mutually tangent.
Find the area bounded by the three circles.
-------
If you mean the area bounded by the tangent points:
Label the centers:
P - 8.5
Q - 11.9
R - 15.7
The lines between the centers form a triangle PQR.
Label the sides p, q & r, side p opposite angle QPR, etc.
---
Find the 3 angles:
Use the Cosine Law:
p^2 = q^2 + r^2 - 2qr*cos(P)
cos(P) = (8.5^2 - 11.9^2 - 15.7^2)/(-2*11.9*15.7) =~ 0.1856233
Angle P =~ 79.3 degs
---
Find a 2nd angle the same way.
The 3rd angle is 180 - sum of the 1st 2.
=================
Find the area of the triangle. Heron's Law will work, or any method.
Subtract the area of each of the 3 sectors.
eg, sector of P = Area of P * (79.3/360)

Answer by advanced_Learner(501) (Show Source): You can put this solution on YOUR website!
this is indeed a classic challenging problem the decimals make it lenghy and hairy ,cant be solved with out drawing,i dont know how to draw three tangent circles but i uploaded an image for you to refer to.

please refer to picture called "circles"
.

first lets find semi-perimeter S of the triangle
semi-perimeter S of the triangle =
=

area of the triangle=
area of the triangle=
area of the triangle=
find angle A ,B and C
NOTE it is not right triangle,use cosine law
Cos A=
Cos A=
Cos A=
A=

Cos B=
Cos B=
Cos B=
B=
cos C=
cos C=

or use the

Cos C=
Cos C=
Cos C=
C= very very minor difference

area sector a=
area sector a=
area sector a=
area sector b=
area sector b=
area sector c=
area sector c=

the area bounded by the three circles=
the area bounded by the three circles=
the area bounded by the three circles=~ is the answer
your feedback is welcome

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