1. Arbelos Area of Arbelos

2. What’s the area of the shaded (grey) region?

4. Diameter = 2R Area of semicircle A =

5. d 1 = 2r 1 d 2 = 2r 2 A1A1 A2A2 Area of semicircle A 1 = Area of semicircle A 2 =

6. A1A1 A2A2 Area of shaded region = Area of arbelo =

7. 2r 1 2r 2 2R As, Area of arbelo = ==

8. 2r 1 2r 2 2r 3 2(r 1 + r 2 ) A1A1 Area of A 1 = A2A2 A 2 = ?

9. A2A2 2r 3 2(r 1 + r 2 ) Area of A 2 ==

10. A1A1 A2A2 Area of shaded region = ==

11. 2r 3 2(r 1 + r 2 ) 2r 4 A1A1 A2A2 A 1 + A 2 = 2(r 1 + r 2 + r 3 ) A3A3 A 3 = ?

12. A3A3 2(r 1 + r 2 + r 3 ) 2r 4 Area of A 3 ==

13. A1A1 A2A2 A3A3 A 1 + A 2 + A 3 ==

14. Pattern Observed 2 small circles, area of arbelo will contain 1 term of r x r y 2r 1 2r 2 r1r1 r2r2 r1 r2r1 r2 Area of shaded region =

15. r1r1 r2r2 r3r3 r1 r2r1 r2 r2 r3r2 r3 r1 r3r1 r3 3 small circles, area will contain 3 terms of r x r y

16. Area of shaded region = r1r1 r2r2 r3r3 r4r4 r1 r3r1 r3 r1 r2r1 r2 r2 r3r2 r3 r1 r4r1 r4 r3 r4r3 r4 r2 r4r2 r4 4 small circles, area will contain 6 terms of r x r y

17. r1r1 r2r2 r3r3 r4r4 r6r6 r5r5 For an arbelo with 6 small circles, it will contain 15 terms of r x r y in its area expression.

18. r1r1 r2r2 r1 r2r1 r2 r1r1 r2r2 r3r3 r1 r2r1 r2 r2 r3r2 r3 r1 r3r1 r3 r1r1 r2r2 r3r3 r4r4 r1 r3r1 r3 r1 r2r1 r2 r2 r3r2 r3 r1 r4r1 r4 r3 r4r3 r4 r2 r4r2 r4 ≡ 2 C 2 = 1 ≡ 3 C 2 = 3 ≡ 4 C 2 = 6

19. r1r1 r2r2 r3r3 r4r4 r6r6 r5r5 ≡ 6 C 2 = 15

20. For an arbelo with n small circles, there will contain n C 2 terms of r x r y in its area expression.