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

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

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##