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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?

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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.

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When the Angles of a Triangle Don't Add up to 180 Degrees

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.

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?