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A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

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Pie Cuts

Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).

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Getting an Angle

How can you make an angle of 60 degrees by folding a sheet of paper twice?

Polygon Cradle

Stage: 3 Short Challenge Level: Challenge Level:1


As $PQRST$ is a regular pentagon, each of its internal angles is $108^{\circ}$. The internal angles of the quadrilateral $PRST$ add up to $360^{\circ}$ and so by symmetry $\angle PRS=\angle RPT=\frac{1}{2}(360^{\circ}-2\times 108 ^{\circ})=72^{\circ}$. Each interior angle of a regular hexagon is $120^{\circ}$, so $\angle PRU=120^{\circ}$.

Therefore $\angle SRU=\angle PRU-\angle PRS=120^{\circ}-72^{\circ}=48^{\circ}$.

This problem is taken from the UKMT Mathematical Challenges.