The shout seems to go out each year, 'Maths is too hard. We can't solve this impossible puzzle." It has happened again this year over the final question in the Edexcel GCSE Maths paper.
The puzzle is to work out the shaded area, given there are 8 regular octagon, with side length 1, arrange as in the image above.
The first step is to recognise the points of the shaded shape are all 90 degrees. This is because the internal angles in a regular octagon are all 135 degree. When two octagons are placed side by side, the external angle is 90 degrees.
The second step is to realise the shaded shape can be made up of a large square and four small triangles
The third is calculating the sides of the squares and triangles. The triangles are easy. The sides adjacent to the right angle are both 1 unit long. This means if we stick two of these triangles together, then the combined area is 1 unit sq. To get the area for the four triangles we double it to get 2 units sq.
The sides of the large square are 1 unit plus the hypotenuse of the small triangle, plus 1 unit. The hypotenuse a is found by using the Pythagorean formula a² =b² + c². This gives a² = 1x1 +1x1, and a as the square root of 2.
The length of the large square is 2 + √2. The area of the square is this (2 + √2)². The total area is (2 + √2)² + 2. Working out the square gives 8 + 4√2.
It was a lovely little puzzle, and it took me far more time to work out how to use HTML to write the expressions than to solve the problem.
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