Question

How do you find the area of a regular octagon given a radius?

Answer 1

Area `= 2sqrt(2)r^2`
where `r` is the radius of the octagon

Explanation:

Consider the diagram below with radius `r`:

A regular octagon can be thought of as being composed of `4` "kite" shaped areas.

The area of a "kite" with diagonals `d` and `w` is
`color(white)("XXX")"Area"_"kite"=(d*w)/2`.
(This is fairly easy to prove if it isn't a formula you already know).

Consider the "kite" `PQCW` in the diagram above.

`/_QCW=pi/2` and `|QC|=|WC|=r`
`color(white)("XXX")rArr |QW|=sqrt(2)r` (Pythagorean)

Therefore (since `|PC|=r`)
`color(white)("XXX")"Area"_"PQCW" = (|PC|*|QW|)/2 = (r*sqrt(2)r)/2 = (sqrt(2)r^2)/2`

The octagon is composed of `4` such kites, so
`color(white)("XXX")"Area"_"octagon" = 2sqrt(2)r^2`