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Three Balls

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

In a Spin

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?


Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Ball Packing

Age 14 to 16
Challenge Level

If a ball is rolled into the corner of a room how far is its centre from the corner?

A sphere of diameter $10$cm is packed into a cubic box whose internal side lengths are $12$cm. To prevent the sphere from rolling around 8 identical smaller spheres are placed in the corners of the box. What is the radius of one of the smaller spheres?