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

| May 7, 2018

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##

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