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?
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).
How can you make an angle of 60 degrees by folding a sheet of paper
Consider the ten trapezia. Add an eleventh trapezium to one edge of the arch.
To move from the left of the arch to the right, each trapezium has to turn the same number of degrees.
Consider the blue edgeof the bottom left hand stone. To complete the arch it has to turn a total of $180$ degrees in 10 steps. That is $18$ degrees for each step, or $9$ degrees for each of the base angles of each trapezium.
Therefore the largest angles of each trapezium are $(90+9)$ degrees and the smallest $(90-9)$ degrees = 81 degrees .
This problem is taken from the UKMT Mathematical Challenges.View the archive of all weekly problems grouped by curriculum topic