2 circles A and B with radius a and b units respectively ( suppose
a < b ) touch at the point P . If circle A rotates x degree at P
anti-clockwisely . ( where 0 < x < 180 )
Find the area of the overlapping region of the 2 circles.
1. Draw a sketch.
2. You'll get 2 sektors whose central angles are 2u and 2w respectively. You know that
$\displaystyle u+w=x$
The area of a segment is calculated by: Area of sector - area of right triangle.
The common side of the two triangles is calculated by:
$\displaystyle a \cdot \sin(u) = b \cdot \sin(w)~\implies~\dfrac ab = \dfrac{\sin(w)}{\sin(u)}$
Unfortunately LaTeX doesn't work anymore. Therefore you wasn't able to see this line:
\tan(w)=\dfrac{a \cdot \sin(x)}{b+a\cdot \cos(x)}
Re-written into the simple syntax it means:
tan(w)=(a * sin(x))/(b+a*cos(x))
3. Calculating the orange area $\displaystyle A_{orange}$:
$\displaystyle A_{orange} = \underbrace{\dfrac{2u}{360^\circ} \cdot \pi a^2}_{sector} - 2 \cdot \underbrace{\dfrac12 \cdot a^2 \cdot \cos(u) \cdot \sin(u)}_{\text{area of right triangle}}$
4. Calculating the green area $\displaystyle A_{green}$:
$\displaystyle A_{green} = \dfrac{2w}{360^\circ} \cdot \pi b^2 - 2 \cdot \dfrac12 \cdot b^2 \cdot \cos(w) \cdot \sin(w)$
5. Add both areas and try to simplify this term a little bit.
Maybe you can use the property: $\displaystyle \sin(u) \cdot \cos(u)=\dfrac12 \left(\sin(2u) \right)$
Thank you very much earboth! Your formula seems OK but can it be
expressed in terms of a , b and x only?
For special values of x :
(1) If x = 0 , the area should be 0.
(2) If x = 180 , the area should be a . a . pi sq.units. (i.e. area of circle A.)
(3) If x = 90 , what will be the area ?
Thanks again !
Hi earboth , after revising some basic rules of trigonometry ,
I get the followings:
Let c be the distance between the centres of the 2 circles,
then c = √a*a+b*b-2ab cos(180-x)
= √a*a+b*b + 2ab cos x
Since a / c = sin w / sin (180 - x)
= sin w / sin x
Thus sin w = a * sin x / c
i.e. w = arc sin ( a * sin x / c )
Similarly u = arc sin (b * sin x / c )
Substituting into your formula ,
Orange area = u / 180 * π * a* a - a * a * sin (2u) / 2
Green area = w / 180 * π * b* b - b * b * sin (2w) / 2
Thus the total area can be expressed as a function of
a , b and x only . ( But I wonder how you get the expression :
tan(w)=(a * sin(x))/(b+a*cos(x))
For a special value of x being 90 ,
the overlapping area will be :
arc sin (b * sin 90 / c )* π * a* a / 180
+ arc sin (a * sin 90 /c )* π * b* b / 180
- the areas of the 2 triangles ( which = ab sq. units )
= arc sin ( b / c ) * π * a* a / 180 + arc sin (a /c )* π * b* b / 180
- ab
= arc sin ( b / √a*a+b*b )* π * a* a / 180
+ arc sin ( a / √a*a+b*b )* π * b* b / 180
- ab
Am I correct ?
Besides to find the overlapping area , we may also find the value
of c ( i.e. the distance between the centres of the 2 circles) given
by the formula : c = √a*a+b*b + 2ab cos x
( 1 ) If x = 0 , cos 0 = 1 , then c = √a*a+b*b + 2ab = a + b
(2) If x = 180 ,cos 180 = - 1, then c = √a*a+b*b - 2ab = b - a
If x is taken randomly from 0 to 180 , what will be the
expected value of c ?