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

1 Answer
Alan P.
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

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