Spheres, cylinders and cones

Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?

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?

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

What 3D shapes occur in nature. How efficiently can you pack these shapes together?

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?

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

A look at different crystal lattice structures, and how they relate to structural properties

A right circular cone is filled with liquid to a depth of half its vertical height. The cone is inverted. How high up the vertical height of the cone will the liquid rise?

A spherical balloon lies inside a wire frame. How much do you need to deflate it to remove it from the frame if it remains a sphere?

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.