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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?

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The Dodecahedron Explained

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

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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.

Mesh

Stage: 5 Challenge Level: Challenge Level:1
Consider the triangle $ABC$ in which all the angles are right angles and in which the edges are quarters of great circles on the original sphere. This is one of the eight congruent 'holes' in the framework through which the escaping spherical balloon can pass.