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Answer: The area of the octagon is 140cm²

The winner is Ian Chantrell of Plymouth, UK. There were 263 entries.

Worked answer

Each angle of the octagon ABCDEFGH must be 135 degrees. The vertices lie on a circle, with centre O. Let angle AOB be θ. Then, as angle ABC is 135°, the reflex angle AOC must be 270° and angle BOC must be 90° – θ. But then similar reasoning makes angle COD equal to θ, etc, and the octagon must look like this, with four sides of length X and four sides of length Y (≥X).

For three of these sides to be used to make a right-angled triangle, its sides must be X, X and Y. Hence Y = X√2.

Also, the area of the triangle = ½X² = 10 and so X = 2√5 (and Y = 2√10).

Now let the area of the octagon be Acm².

The corner triangle shown dotted above has area ¼ X². So the area of the square obtained by adding the four corner triangles is A + X² = A + 20.

But this square has side Y + 2.X/√2 = 2√10 + 2.2√5/√2 = 4√10.

So the area of the square is 160cm², and it follows that A = 140.

Alternatively (and more swiftly) note that the octagon must in fact be made up of 14 of the right-angled triangles.

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