Pretty rings all in a row

It’s the Olympics, so how about a puzzle that gets you to look more closely at the famous symbol of the Games, the five circles?

As you see, the circles are linked, pairwise. Each link produces a small area that’s common to the two linked circles. For example, the blue and yellow circles share a common area between points A and B on their circumferences (ie, A and B are the two points where the blue and yellow circles intersect). There are four such shared spaces: blue and yellow, yellow and black, black and green, green and red.

Assume each circle has a diameter of 2 cm. Assume that if you draw a straight line from A to B, its length is 1 cm. Assume the shared spaces are all equal in area.

Question: What is the total area of the four shared spaces? (All you need is elementary geometry and arithmetic).

Hint: Note that the area of each circle is πr2, or π times the square of its radius. Since its diameter is 2, the radius is 1 cm; thus the area of each circle is πcm2.

Call the centre of the blue circle O.

Draw lines from O to A (OA), O to B (OB), and A to B (AB). OA and OB are both radii of the blue circle, thus they are 1 cm long. So is AB. Thus the triangle OAB is an equilateral triangle, so each of its angles is 60º; in particular, the angle at point O is 60º. Since a circle is 360º all the way around, you can draw six such equilateral triangles in the blue circle, all equal, all radiating from point O.

That divides the circle into six equal slices: the one defined by points O, A and B, and five more. Thus the area of slice OAB is one sixth of the area of the circle, thus π/6 cm2.

What’s the area of the triangle OAB? Draw a line from the midpoint of AB, call it C, to O. AC = 1/2 cm, OA = 1 cm, and the angle at C (angle OCA) is 90º. Thus by Pythagoras, OC = √3/2 cm.

Nearly done! You now have all you need to calculate the area of triangle OAB; then the outer part of slice OAB, ie, outside the line AB but within the blue circle. The rest is simple.

For the answer, see bottom of the page