C2: Radian Measure

Two circles C1 and C2, both of radius 12 cm, have centres O1 and O2 respectively. O1 lies on the circumference of C2; O2 lies one the circumference of C1. The circles intersect at A and B, and enclose the region R.

(a) Show that Angle AO1B = 2/3 pi radians.

(b) Hence write down, in terms of pi, the perimeter of R.

(c) Find the area of R, giving your answer to 3 s.f..


Would really appreciate clear working; I'm really stuck.

Thanks!

This is the method:

Draw a clear sketch and join AO1, AO2, BO1, BO2, O1O2.

All of AO1, AO2, BO1, BO2, O1O2 are radii so the 2 triangles are equilateral, all internal acute angles being therefore pi/3. So AO1B = 2/3 pi radians.

Length of arc AB:

since AO2B = 2pi/3 that is a third of 2pi, then length of arc AB is a third of the circumference of either circle. R perimeter is twice this.

Use the formula 0.5 r² ? (with ? = pi/3) to get the area of the sector O1AO2, then subtract the area of one of the equilateral triangles to get the area of one of the small segments. Total the area of 2 of the equilateral triangles and 4 of the segments to get R.

Aitch