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Major Trapezium

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A small circle is placed inside a square just touching the sides. The square is on top of a big circle with its vertices on the circle. Finally the big circle is on top of a trapezium just touching the parallel sides. Using the measurements and clue given, find the area of the trapezium.

Clue: The small circle has a circumference of 12 centimetres.


Here is a solution by Abbas, from Jack Hunt School, Peterborough. Congratulations Abbas, this is a very neat solution.

The area of the trapezium is half the sum of the parallel sides times the vertical height.

The vertical height of the trapezium is the diameter of the large circle.

From the clue given, the radius of the small circle is $6 \over \pi$.

Using this value and by Pythagoras theorem, the radius of the large circle is \[\sqrt {72}\over \pi\].

Therefore, the diameter of the large circle is $$2\sqrt {72}\over \pi.$$

Hence, the area of the trapezium is (2+5) divided by 2 multiplied by the diameter of the large circle which equals 18.907 square centimetres (to three decimal places).

Well done Danniella of the Mount School York for spotting that the diagram as originally drawn was impossible though this did not make any difference to the solution. Good solutions were also sent in by Laura , Rosie , Jenny , Catherine and Kayley of Davison High School, Worthing; Arti , Ruoyi, Lizzie and Shelli at the North London Collegiate School (NCLS) Maths Puzzle Club; Alicia, Nisha , Suzanne Abbott and Christiane of the Mount School, York and Hereward of Goldington Middle School, Bedford. Well done all of you!