Here 3 circles are touching each other.
Now how can one find the area of the blue shaded region in the given picture?
[Math] The area of the region bounded by three mutually-tangent circles
geometry
Related Solutions
Let's get a figure in here, eh?
We can exploit the fact that the points of mutual tangency of the three inner circles (red points in the figure) form an equilateral triangle; we also know that the red points are the midpoints of the segments formed by joining any two of the three blue points (the centers of the inner circles).
We then deduce that the triangle formed by the red points is half the scale of of the triangle formed by the blue points, and find that the equilateral triangle formed by the blue points has a side length of 6 ft., and that the inner circles have a radius of 3 ft. Using the law of cosines, we reckon that the distance from a blue point to the center of the triangle formed by the blue points is $2\sqrt{3}$ ft. Adding to that the radius of an inner circle, we find that the radius of the outer circle is $3+2\sqrt{3}$ ft.
The area of the outer circle is $\approx$ 131.27 square feet.
If you know about polar coordinates (and especially equations of circles that are tangent to the origin) and you know about integration, then you could find these areas by placing the origin at $G$. You would make use of the radii ($3$ and $9$) to find the angles at which the $A$- and $B$-centered circles are oriented relative to your primary polar axis.
Best Answer
Hints have been given in the comments. To make it more complete, I add a few more.
(1) $AB = R_1 + R_2$, and similar results for $BC$ and $CA$.
(2) Because all sides of $\triangle ABC$ are known, all three angles (in radians) can be found by applying cosine law (three times) (or cosine law then sine law and then " angle sum of triangle").
(3) Need to apply the conversion ratio ($\pi$ radian $= 180$ degrees) if not already done so in (2).
(4) Area of the yellow sector $= \frac {1}{2}(R_2)^2 \theta $. Again, $\theta$ should be in radian.
(5) Area of $\triangle ABC$ can be found by Heron’s formula or by the area formula $A = \frac {1}{2}ab. \sin C$.