First let’s figure out the radius of the semicircles. Lines from the ends of the chord to the centre of the circle will bisect the angles of the equilateral triangles, forming 30 degrees angles. From this we can work out that the perpendicular distance from the centre of the unit circle to the chord is 1/2.

We can now set up a Pythagorean triangle with the radius R of the semicircles as an unknown.

(1-R)^2 = R^2 + 1/4

1 - 2R + R^2 = R^2 + 1/4

1 - 2R = 1/4

3/4 = 2R

R = 3/8

If the distance from the centre to the chord is 1/2, then so too is the distance from the chord to the top of the circle. Again we can set up a Pythagorean triangle, this time with the radius of the small circle as the unknown:

(3/8+r)^2 = (3/8)^2 + (1/2-r)^2

(3/8)^2 + 3r/4 + r^2 = (3/8)^2 + 1/4 – r +r^2

3r/4 = 1/4 - r

7r/4 = 1/4

r = 1/7