Lengths of diagonals of a regular octagon

Geometry Level 3

Regular octagon $ABCDEFGH$ has side length of $2$ . Find $AG \times AE$ .

Note:

“ $\times$ ” means multiplication.

\dfrac{64\sqrt{2}-64}{8}

8\sqrt{2}+8

2\sqrt{2}+4

\dfrac{16\sqrt{2}}{4-\sqrt{2}}

1 solution

A Former Brilliant Member
Feb 15, 2020

$\angle {AHG}=135\degree$ and hence $|\overline {AG}|=4\sin {67.5\degree}$ . $\angle {AGE}=90\degree$ and hence $|\overline {AE}|=4√2\sin {67.5\degree}$ . So $|\overline {AG}|\times |\overline {AE}|=16√2\sin^2 {67.5\degree}=8√2(1-\cos {135\degree})=8√2(1+\dfrac{1}{√2})=\boxed {8√2+8}$

Thanks for posting a solution.

A Former Brilliant Member - 1 year, 3 months ago

Lengths of diagonals of a regular octagon

1 solution

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