article
The dodecahedron explained
What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?
Spheres, cylinders and cones
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articleVolume of a pyramid and a cone
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts. -
articleMouhefanggai
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. -
articleWhen 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. -
articleConic sections
The interplay between the two and three dimensional Euclidean geometry of conic sections is explored in this article. Suitable for students from 16+, teachers and parents. -
articleCurvature of surfaces
How do we measure curvature? Find out about curvature on soccer and rugby balls and on surfaces of negative curvature like banana skins. -
articlePaint rollers for frieze patterns
Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.
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problem2D-3D
Age16 to 18Challenge level
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?
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problemConical bottle
Age14 to 16Challenge level
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? -
problemMesh
Age16 to 18Challenge 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?