Spheres, cylinders and cones

problem
There and back again

Age

11 to 14

Challenge level

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

Age

16 to 18

Challenge level

Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere.
problem
Pythagoras on a sphere

Age

16 to 18

Challenge level

Prove Pythagoras' Theorem for right-angled spherical triangles.
problem
Flight path

Age

16 to 18

Challenge level

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.
problem
Peeling the apple or the cone that lost its head

Age

14 to 16

Challenge level

How much peel does an apple have?
problem
Tin tight

Age

14 to 16

Challenge level

What's the most efficient proportion for a 1 litre tin of paint?
problem
Efficient cutting

Age

11 to 14

Challenge level

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.
problem
Air routes

Age

16 to 18

Challenge level

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.
problem
Mesh

Age

16 to 18

Challenge level

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?
problem
Sponge sections

Age

7 to 11

Challenge level

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.