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2D-3D
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?
The Dodecahedron Explained
What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?
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.
Age 16 to 18
Challenge Level
You only need elementary trigonometry to do this question if you use the guidance given in the hints.
London is at longitude $0^o$ and latitude $51.5^o$ North and Sydney at longitude $151^o$ East and latitude $34^o$ South. Taking the earth to be a sphere with radius 6378 kilometres, calculate the distance between London and Sydney.
If the flight path is the shortest route at an altitude of 6 kilometres calculate the distance along the flight path.
NRICH is part of the family of activities in the Millennium Mathematics Project.