Three circles are drawn tangentially to each other, their centres collinear as shown. The line AB is tangential to the two smaller circles and is 8 units long.

Philip Morgan, age 17, Judd School, Tonbridge, Kent, Zoë Hayward, age 16, Outwood Grange School, Wakefield and Sue Liu of Madras College, St Andrews and Katharina Jurges, age 17, of the Lycee International des Pontonniers, Strasbourg, France sent solutions. Well done!

Let the radius of the big circle equal a, the radius of the medium circle equal b and the radius of the smallest circle equal c. Considering the diameter of the big circle: 2a = 2b + 2c, so a = b + c.

Let the centre of the larger circle be O and the point where the chord touches the smaller circles be D.

Construct the right-angled triangle AOD. Now, by Pythagoras Theorem,

AO2 = OD2 + DA2 (b + c)2 = (b − c)2 + 42 bc = 4

The area needed is π(b + c)² − πb² − πc² = 2πbc = 8π.