Either of two unit circles passes through the center of the other. How do you prove that the common area is 23π32 areal units?

2 Answers

Proved in the explanation

Explanation:

The common area is enclosed by unit circle arcs subtending

120o=23π radian, at the respective centers.

For this arc, sector area of a unit circle

=23π2π(area of unit circle)=(2/3pi)/(2pi)(pi)=pi/3#

One half of the common area

= this sector area less area of the inner triangle, of sides [1,3,1]

=π3(12)(3)(12)

Twice this is the common area =23π32 areal units.

I welcome a graphical depiction,.from an interested reader.

enter image source here

Sep 20, 2016

see explanation.

Explanation:

enter image source here

1) Equilateral triangle area Ae=34×12=34

2) yellow area Ay=π634=2π3312

3) Common area Ac=2Ae+4Ay

Ac=2×34+4(2π3312)

=32+2π333

=33+4π636

=4π336=2π332