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

1 Answer
Jan 5, 2016

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

Explanation:

Consider the diagram below with radius #r#:
enter image source here

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#