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How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square? This gives a short summary of the properties and theorems of cyclic quadrilaterals and links to some practical examples to be found elsewhere on the site.

An Equilateral Triangular Problem

Age 11 to 14 Challenge Level:

Well done to Book and Priya from Bangkok Patana School in Thailand, who sent us their work on this problem:

When moving around some of the smaller shapes cut out from the equilateral triangle, we actually found many solutions for making trapezia using trial and error, but also remembered to include irregular trapezia. In total, we made 15 trapezia!

To start off, you should write down all the angles in the small shapes, since it makes it easier to calculate the angles to check if you actually made a trapezium. Look in the equilateral triangle diagram and see if you can spot any possible trapezia from just looking. After that, try adding more triangles to it to make new ones but be sure to check if the total of the interior angles adds up to $360^\circ$. Book and Priya were able to find the angles in each of the cut out shapes and they were able to make 15 different trapezia.

However, there is still more to look at in this problem so we have added it to our Toughnuts:

• If the area of the smallest equilateral triangle is one unit, what is the area of each of the other shapes?
• How many different parallelograms (which are not rectangles) can you make?
• How many different rectangles can you make?
• Which other quadrilaterals can you make?
Do make sure that your angles add up to the right amount, and when you put shapes together, make sure that the joined sides have the same length.