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A property of Regular Octagons

Prove that the area of a regular octagon is the product of its longest diagonal and its shortest diagonal


There’s an easier solution by cutting & pasting cleverly transforming the octagon into the area of a rectangle formed by cutting and pasting (where the length & width of this rectangle are the longest & shortest diagonals of this regular octagon) and the claim follows.

But, let’s look at a more “analytical way” though more complicated than the synthetic approach mentioned above.

The details are shown in the images (in order) beginning with the background facts/results needed prior & how to use it to formulate the solution accordingly:

screenshot_2016-11-02-13-36-33 screenshot_2016-11-02-13-36-52 screenshot_2016-11-02-13-37-09 dsc_0005 dsc_0006 dsc_0007

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