The centres of three circles that touch each other externally form a triangle of sides 21 cm, 22 cm and 23 cm. Find the approximate area of the smallest circle?

Let the sides of the triangle be AB, BC, and AC.

Each of the sides is made of radii of two circles. Let us assume that the radius of the three circles are x, y, and z.

Hence, 

 x+y= 21 ..............(1)

 y+z= 22 ...............(2)

 z+x= 23 ................(3)

Now from equation 3, z= 23-x ........... (4)

Substituting equation 4 in equation 2, we get x-y= 1 ...............(5)

Using equation 1 and 5, we get x= 11, y= 10, and z= 12.

The smallest radius is 10 cm, and the area of that circle is $$\pi10^2\ $$, which is approximately 314$$cm^2\ $$