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Age 16 to 18

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A wire frame is made by wrapping a wire around the 'equator' of a perfectly spherical balloon of unit radius and wrapping two wires at right angles around the sphere going through the North and South poles. The balloon is then deflated but remains perfectly spherical until it can just be moved out of the framework. What is the radius of the largest sphere that can pass through this frame, without distorting it, from inside to outside?

Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of equal volume with the centre of each sphere on the surface of the other. What is the volume of intersection?

What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?

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.